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= 0.3;Young’s modulus of the FRP Ep |
= 200 GPa; Poisson ratio of the FRP νp |
= 0.3;Young’s modulus of the adhesive Ea |
= 6.7 GPa; Poisson ratio of the adhesive νa |
= 0.3.The external concentrated load P |
= 100 × 103 |
N.It is assumed that there are initial cracks of small lengths, 2 mm for instance, on the interface between the concrete and adhesive layer near the end of the FRP plate as shown in . Half of the beam is considered due to its symmetry. In the adhesive layer and FRP layer, four layers of elements with nearly square shapes are arranged. The refined mesh is also arranged near the end of FRP plate and the tip of initial cracks. The minimum size of elements around the tip of initial cracks is about 0.1 mm. The whole finite element mesh is shown in , and the local finite element around the tip of the initial cracks is shown in . Plane stress model and 8-node quadrilateral element are employed. Two layers of elements are arranged in the thickness of the adhesive layer.For consideration of stress singularity at the crack tip, the singularity elements are arranged in the domain round the crack tip. A singularity element is produced by moving the node of mid-sides of the element to 1/4 of side length toward the tip. The singularity element can produce singularity of r−1/2 at the crack tip ( shows the element arrangement around the crack tip and generation of the singularity elements. where the relations of the energy release rate and crack length are shown. It is seen from the figure that crack growth has three phases, e.g. cracking initiation, stable growing of the crack and unstable cracking. When the crack length is very short, approximately 2–6 mm, the strain energy release rate increases very quickly as the crack grows. It is shown that a small initial interfacial crack may cause the delamination of the interface between the FRP-concrete. As the crack length becomes large, energy release rate will change slowly and a ‘plateau range’ appears in the curve as shown in . This is a stable extension phase of the crack. The energy release rate will quickly increase when the crack grows continuously, which implies that the crack will unstably grow up to the complete failure of the interface.It is noted that each phase of the crack growth is influenced by the geometry and property parameters of the hybrid beam. To prevent the delamination of the interface, the corresponding strategies should been applied in the design of the hybrid beams, for example, suitable selection of the materials, control of bonding properties of the interface. A quantitative analysis of effect of the parameters on the interfacial cracking is given below.To investigate different influences of the parameters of the concrete, FRP and adhesive on the interfacial fracture can provide quantitative images to the engineers. In order to consider the effects of the structural parameters on the interfacial fracture behavior, the finite element results of the strain energy release rate for different property and geometry parameters of hybrid beam are demonstrated in this section. The varied material parameters are Young’s modulus of concrete, FRP composite and adhesive layer. The varied geometry parameters are thickness of FRP plate and adhesive layer. We limited our attention to the stable growth phase of the crack. Accordingly, crack extension length less than about 100 mm is considered. shows the curves of the energy release rate versus crack length for different values of the Young’s moduli of the concrete. Here three sets of the Young’s moduli of the concrete (Ec |
= 20, 30 and 50 MPa) are considered. It is shown that the Young’s modulus of concrete influences the interfacial cracking considerably. The higher stiffness of the concrete leads to lower energy release rate of the interfacial cracks. It implies that for a stiffer beam strengthened by a relatively softer sheet, the interfacial initial cracks may not bring the delamination of the interface. However, the Young’s modulus of FRP in practice is several times over one of the concrete. Consequentially, the delamination of the interface is one of the failure modes of the hybrid beams.The influence of the stiffness of FRP on the energy release rate is shown in where three sets of Young’s moduli of the FRP, Ep |
= 50, 100 and 200 MPa, are accounted for. Actually, the stiffness of FRP is in range of 40–150 GPa, depending on the properties of the fiber and matrix of the FRP. The data in larger range (50–200 MPa) is considered here to cover the possible properties of the FRP. It is shown that the strain energy release rate is sensitive to the Young’s modulus of the FRP sheet. When the stiffness of the FRP increases, the strain energy release rate of the interfacial crack obviously increases. In fact, the stiffness mismatch between the FRP and concrete is a primary reason for the debonding of the interface. shows the influence of the different stiffness of adhesive layer on the energy release rate, where three sets of Young’s moduli of the adhesive are Ea |
= 3, 4 and 6.7 MPa. The stiffness of the adhesive layer hardly affects the strain energy release rate of the crack in the initial cracking phase. But the effect becomes obvious for growth of the crack in the second or final phases. Because the adhesive layer has small stiffness and small volume fraction in the hybrid beam, the effect can be omitted.The strain energy release rates are calculated in order to account for the effects of the thickness of the FRP plate and adhesive layer, respectively. The relations between the strain energy release rate and crack length are plotted in . It is seen that the larger thickness of the FRP leads to higher level of the energy release rate. Recalling the influence of the Young’s modulus of the FRP, we can conclude that the tension stiffness (EpAp) or the bending stiffness (EpIp) is an important parameter for the cracking of the interface in hybrid beam. Although the bigger stiffness of the FRP may enhance the overall strength of the hybrid considerably, it will contribute to the delamination of the interface. This implies that there is an optimal value of stiffness ratio of concrete and FRP plate. shows the influence of the thickness of the adhesive layer on the strain energy release rate. Like the situation in , the thickness of the adhesive layer slightly affects the initial cracking but it becomes obvious with the growth of the crack. In engineering design reasonable thickness of adhesive layer is essential to enhance the performance of the hybrid FRP-concrete beams.The cracking processes of the interfacial cracks in FRP-concrete beams were analyzed by finite element method. The virtual crack extension method was used calculate the energy release rate of interfacial cracks. The influences of the material and geometry parameters on the interfacial fracture were investigated. The research can be concluded as follows:A small initial interfacial crack may cause the debonding of the interface between the FRP and concrete.The growth of the interfacial crack has three phases, e.g. the cracking initiation induced by the small initial crack, stable growth of the crack and unstable growth which causes the complete delamination of the interface.The parameters including the stiffness and the thickness of the FRP and the stiffness of the concrete influence dramatically the cracking of the interface. The bigger stiffness mismatch between the FRP and concrete results in easy debonding of the interface in the hybrid beam.The material and geometry parameters of the adhesive layer cause slight effects on the strain energy release rate of the interfacial crack.Cause and mitigation of dilatancy in cement powder pasteSuperplasticiser (SP) becomes essential in contemporary concrete manufacturing because it decreases the water demand for a prescribed set of concurrent strength and workability requirement. On the other hand, SP also creates unfavourable dilatancy (i.e. shear thickening) that decreases the uniformity of mixing and pumpable distance of concrete, as well as creates difficulty in in-situ handling. Although it is known that the dilatancy is created by clustering of free and adsorbed polymers of SP, the correlation of them is not fully understood. Herein, the dilatancy of cement powder paste was studied using a co-axial rheometer. It is evident that with the addition of SP, dilatancy of paste increases initially from zero up to certain dosage. Further addition of SP decreases dilatancy as it effectively disperses the cement particles. At a given SP dosage, replacing cement partially with pozzolanic filler such as fly ash or silica fume can effectively decrease dilatancy depending on SP dosage. It is because the fillers can fill up the interstitial void of the cement particles and free up more water for cement hydration to produce Ca2+ via ettringite formation. Hence, it attracts more negatively charged SP to adsorb to the cement surface, and decreases the free SP polymers that cause dilatancy.Third generation superplasticiser (SP) which is polycarboxylate-based becomes crucial in the manufacturing of high-performance concrete. This type of SP can decrease fine particles’ flocculation and increase excess water for workability. Alternatively, for a given workability requirement, less water is needed and strength is increased The influence of SP on concrete segregation is understood To study the dilatancy of cement powder paste or concrete, its plastic viscosity needs to be measured. In industrial application, they are measured by simplified empirical approach known as slump flow and V-funnel. These methods of measurement can only tell us the flow characteristic at low shear rate under gravity effect only. Nonetheless, studying of flow characteristics of paste and concrete at moderate or high shear rate is equally important. This normally refers to the process when large work done on concrete is carried out to make it workable – mixing, pumping and placing. In mixing, fresh concrete will be subjected to 10–60 s−1 shear rate From the concrete workability’s point of view, concrete flow should not be too viscous in any shear rate To mitigate the effect of dilatancy, the working mechanism of SP and its effects on rheology of cement paste need to be understood. Yamada et al. Apart from the studies carried out on the adsorption surface mechanism, the effect of poly-carboxylate SP on cement paste with other s or fillers has also been researched. Nagataki et al. Effects of time and temperature were studied by Kong et al. It is evident from the above that when SP is added to the cement and/or other fine particles, the polymer chains will adsorb to the surface of fine particles. However, the adsorption affinity of SP is not the same on the surface of various particles General Purpose Portland Cement (GP), which complies with AS 3972 . SG is the specific gravity and SSA is the specific surface area (m2/kg). Their particle size distributions (PSD) are shown in , it is evident that FA is slightly finer than cement, whereas SF is much finer than cement.All dry powders were firstly prepared and then mixed in a table-mounted mortar mixer as shown in for at least a minute. It consists of a bowl and beater easily fitted and removed that mixes the powders with planetary mixing action. Subsequently, three quarters of the designed water were introduced in the mixture and then mixed for 3 min. The second half of the ingredients and remaining water were then added and mixed for another 3 min. Next, SP (if any) was added and the last mixing process will last for about 5 min. For cement paste with SF, the mixing time was longer which could be extended to 7 min because of its ultra-fineness. The whole mixing process last no>15 min in general. Lastly, the mixture will be put into a cup. A co-axial rheometer as shown in shows a co-axial rheometer that can measure the torque (T) that the cement powder paste was experiencing at certain pre-set angular velocity (Ω). The rheometer consisted of a cup and a probe. The inner diameter of the cup was 155 mm while the probe had a diameter of 95 mm. The probe was put in the sample in the test and installed with sensors for measuring the torque required to keep it stationary. During testing, the outer cup rotated at different velocities prescribed by the user. When it rotated, the sample in the annulus would drag the probe to rotate, while the probe always remained stationary.To start with, the probe was inserted into the cylindrical cup containing the test sample concentrically. Then a prescribed loading programme was set into the computer to rotate the cylindrical cup at certain range of angular velocities (Ω). It increased from 5 to 100 RPM at 5 RPM interval, with each interval maintained for 15 s. After rotating at 100 RPM for 15 s, the angular velocity went down to 5 RPM in 5 RPM interval with each interval maintained for 15 s. The first cycle ensured that the entire mixture had the same shearing history. Then, data acquisition started in the second and third cycles, in which the decreasing shearing rate branch were recorded The VISKOMAT XL only recorded the torque and angular velocity variation during the test, but not the required shear rate (γ̇) and shear stress (τ), which are essential to obtain the yield stress, plastic viscosity and the extent of dilatancy of the tested sample. The conversion from torque to shear stress was quite straightforward, however, that from angular velocity to shear rate was the reverse. The commonly adopted narrow gap where γ̇ is the shear rate at the probe surface, s is the diameter ratio of the outer cup to the inner probe, Ω is the angular velocity (s−1), t is a dummy variable, τ is the shear stress at the inner probe given by Eq. where T is the measured torque, r is the radial distance from the centre of rotation and L is the length of the probe. In this study, as well as most of the other studies on cementitious paste, fairly linear relationship was obtained for ln Ω plotted against ln τ. Then N' in Eq. is insignificant. Since the function f(t) is bounded, the second term in the square bracket in Eq. becomes higher-order and can be ignored.Having obtained the required shear stress and rate, they were fit into appropriate rheological model to obtain the parameters. Without SP, linearly Bingham Model where τ and γ̇ are the same as before. τo, μ, c and k are to be determined by multi-variables nonlinear regression analysis. With Eqs. determined, τo is the yield stress and the tangent slope is the plastic viscosity. The extent of dilatancy is represented by c/μ in Eq. . When c/μ (or n) is zero (or unity), it represents a linear Bingham fluid. When c/μ (or n) is smaller/larger than zero (or unity), it represents a shear thinning/thickening fluid.Wet method measurement of packing density of the tested sample as recommended by Wong and Kwan Four groups of cement powder paste sample containing a total of 42 mixes were prepared and their details summarised in . In all mixes, the volume ratio of cement (and FA/SF, if any) to total volume of paste was 0.444. Hence, the water-to-total fine powder volumetric ratio for all mixes was 1.25 (or ~0.40 by weight). Readers should note that the adopted water ratio corresponds to the cement powder paste in typical normal- to high-strength concrete but not ultra-high-strength concrete (i.e. >100 MPa). The following summarises the details of mixes in each group:Group 1: The paste in this group contained no SP. The first sample consisted of cement and water only. Subsequent samples in this group had partial volume of cement replaced by either FA or SF. The replacement ratios were 15%, 30% and 45% by volume for FA, and 5%, 10% and 15% for SF.Group 2: The paste in this group contained cement, water and SP only. There were 5 mixes with SP dosage varying from 0.16% (Sample B) to 1.0% (Sample F) by weight of cement.Group 3: The paste in this group contained cement, FA, water and SP. The percentages of FA by volume of total powder were 15%, 30% and 45%. The weight of SP varied from 0.16% to 1.0% of powder’s weight for all percentages of FA.Group 4: The paste in this group contained cement, SF, water and SP. The percentages of SF by volume of total powder were 5%, 10% and 15%. The weight of SP varied from 0.16% to 1.0% of powder’s weight for all percentages of SF. shows the graph plotting shear stress against shear rate obtained for some selected Group 1 samples without SP. From the figure, it is seen that cement powder paste with no SP added is linearly plastic disregarded of the composition of powder. The determined coefficients of all Group 1 samples as shown in was obtained by fitting the results using linear Bingham Model. shows the rheological parameters obtained for all mixes in Group 1. It is seen that the yield stress decreased as FA was used to replace cement with equal volume (see FA-15A, FA-30A & FA-45A). When FA was added, the yield stress decreased from 249 Pa to 107 Pa (i.e. 59% decrease) as FA increased from 0 to 45% by volume of powder. It happened because FA was more spherical than cement, which decreased the internal friction of powders by ball-bearing effect when it flowed/deformed. On the other hand, when SF was added to replace partial cement, the yield stress did not change significantly at 5 and 10% replacement, which were at 258 and 236 Pa respectively. It decreased more significantly to 187 Pa when 15% of SF was added. Under no SP, there was serious agglomeration of the cement and the ultra-fine SF particles. Therefore, filling of ultra-fine SF particles into the cement’s interstitial void was ineffective. The filling effect improved slightly as the volume/weight of SF increased solely because of the higher probability for the SF particles to fill into the interstitial void.The plastic viscosity decreased as FA replaced equal volume of cement. It decreased from 0.866 to 0.086 when the volumetric ratio of FA increased from 0 to 15%. The viscosity then increased slightly from 0.086 to 0.216 Pas when FA increased from 15 to 45%. Initial addition of spherical FA to partially replace angular cement particles could improve the packing of powder and flowability. Nonetheless, subsequent addition of FA could cause more loosening effect than filling effect (i.e. remaining interstitial void after filling some FA was too small for FA to keep filling in), thus the void increased subsequently and trapped more water. The optimal percentage of FA replacement considering the plastic viscosity was between 15 and 30%. Similar observation was obtained when SF was added to replace cement with identical volume, the plastic viscosity reduced initially from 0.606 to 0.064 Pas when the volumetric ratio of SF increased from 0 to 5% of powder. Then the plastic viscosity increased slightly from 0.064 to 1.089 Pas when SF increased from 5% to 10% owing to the increased contact surface area of ultra-fine SF particles. The optimal percentage of SF replacement considering viscosity was between 5 and 10%. show the graphs plotting shear stress against shear rate obtained for some selected Groups 2 to 4 samples respectively containing SP. Again, shows the determined coefficients obtained by fitting the acquired test results into the Modified Bingham (Eq. lists the rheological parameters obtained for all mixes in these groups. Unlike the graphs in are non-linear, which indicate that the plastic viscosity of cement powder paste containing SP increased as shear rate increased – dilatancy. The yield stress for these groups of paste sample is given by τo while the plastic viscosity is not a constant as it increased as shear rate increased. In this study, the authors investigate the variation of viscosity just at certain shear rates which are meaningful to the practical use of concrete in real application. During casting, the maximum shear rate of concrete is about 10 s−1. At last, the extent of dilatancy can obtained by the ratio of plastic viscosities at 30 s−1 to 10 s−1 or c/μ or n in respectively the Modified Bingham or Herschel-Bulkley Model.Comparing the value of c/μ or n shown in , it is evident that SP creates dilatancy. Interestingly, it was observed that the effect of dilatancy was not directly proportional to the SP dosage. Initially when small dosage of SP was added to the paste, dilatancy occurred because of the clustering of the free and adsorbed SP polymer chains The yield stress of the tested paste samples in Groups 2–3 decreased as the SP dosage increased because of the better dispersion of particles, increase in wet packing density and excess water. The excess water acts as a lubricant by forming a thin film on the surface of fine particles – named “Water Film Thickness” The effect on dilatancy by cement replacement ratio of FA or SF is now explained. At an SP dosage of 0.5% (i.e. Sample D) as shown in , the dilatancy of paste as represented by c/µ or n decreased as FA increased from 0 to 30%, after which it increased slightly. It is worth noting that the SP dosage and water-to-powder ratio did not change in Sample D, the diminished dilatancy must have been caused by the FA replacement of cement. The initial reduction of dilatancy when FA replaced cement up to 30% was because of the better packing of cement powder paste as explained before. The slightly increase of dilatancy at FA = 45% was merely because of the inadequate SP (=0.5%) to disperse effectively the fine particles for improving packing density. It can be verified by that fact that at a higher SP = 1.0% (i.e. Sample F), the dilatancy decreased monotonically from FA = 0 to 45%. On the other hand, the dilatancy of Sample D with SP = 0.5% decreased as SF increased from 0 to 10%, after which dilatancy increased slightly. The initial decrease of dilatancy can be explained by the same theory as that of FA. However, it cannot be concluded whether the increase of dilatancy beyond SF = 10% was because of inadequate SP as the same trend was also observed at higher dosage of SP = 1.0% (i.e. Sample F). Further tests on the same SF paste samples containing SP > 1.0% will be required for verification. For clarity, the dilatancy at SP = 0.5% represented by c/μ is plotted in against the replacement percentage of FA or SF.It is generally accepted that dilatancy is caused by disorderly movement of particle during fluid flow. In cement paste and concrete, it is about the formation of irregular clusters that obstruct the flow at high shear rate. Before SP was used as a chemical admixture in producing concrete or mortar, there was no evidence showing that the hydration of cement will cause disorderly obstructing flow. When SP becomes popularly used as chemical additives in making concrete, researchers were aware of the dilatant behaviour. The following recite some of the previous findings on dilatancy of cement and concrete.From the above discussion, since clustering always occurs at the interstitial void created by the free polymer chains, the most effective way of mitigating dilatancy will be to decrease the amount of free SP polymers. This can be effectively achieved by: (1) Decreasing the interstitial void; (2) Enhancing cement hydration to produce more Ca2+ on the surface of cement particles for free SP polymers to adsorb. Increasing the wet packing density can achieve these objectives simultaneously in cement paste , the initial addition of SP increases the dilatancy. Upon further addition of SP, the dilatancy decreases. It is because the wet packing density increases as SP increases . The packing density also increases initially below the threshold SP dosage but it did not decrease the dilatancy because the amount of SP polymers provided is too low to fully disperse the fine particles resulting in large interstitial void and poor cement hydration. The minimum threshold SP dosage varies with the content of cement powder paste. In Groups 2 and 3, it occurs at about 0.33%, while in Group 4, it increases to 0.65%. This is because in Group 4 where there was ultra-fine sized SF in the paste samples, more SP is required to fully disperse the finer particles due to the increase in surface area of fine particles. And since the size of cement and FA is similar, the quantity of SP requires to fully disperse these fine particles will be similar.The dilatancy of various cement powder pastes containing pozzolanic filler (i.e. fly ash FA and silica fume SF) or not (i.e. pure cement paste) was investigated by a co-axial rheometer in this study. The tested samples presented in this paper were divided into four groups that had various combinations of powder content and polycarboxylate superplasticiser (SP) dosages: (1) Group 1 contained cement powder paste without SP; (2) Groups 2 to 4 contained cement powder pastes with SP without or with FA/SF. The percentage of FA varied from 15 to 45% by volume of powder, whereas SF is from 5 to 15%. The SP dosage was measured by weight of powder and it varied from 0.15 to 1.0%. The following summarises the crucial findings:Cement powder paste contains no third generation polycarboxylate based SP behaved as a linearly plastic fluid with no dilatancy, which followed linear Bingham Model.All other cement powder paste containing same type of SP exhibited dilatancy, which followed Modified Bingham or Herschel-Bulkley Model.Dilatancy of cement powder paste was caused by the clustering of adsorbed and free (i.e. non-adsorbed) SP polymer chains since the adsorbed SP polymers could not form clusters with other adsorbed polymers due to electrostatic repulsion.Dilatancy of cement powder paste increased initially when SP was added because of the increased polymer chains. Having reached a threshold SP dosage of about 0.33–0.65%, the dilatancy decreased as SP dosage increased because of better dispersion of particles.Dilatancy of cement powder paste diminished as FA increased up to 30% at SP = 0.5%, whereas it decreased as FA increased monotonically at SP = 1.0. For cement powder containing SF, the dilatancy increased monotonically as SF at both SP = 0.5% and 1.0%.Dilatancy of cement powder paste could be negatively correlated to its wet packing density. Increasing wet packing density could decrease the free SP polymers by: (i) decreasing the interstitial void, (ii) enhancing cement hydration for better adsorption of free SP polymers.The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.Supplementary data to this article can be found online at The following are the Supplementary data to this article:Controlling multi-function of biomaterials interfaces based on multiple and competing adsorption of functional proteinsA multifunctional biomaterial interface is created based on competing adsorption of BMP-2 and fibronectin on a polychloro-para-xylylene (PPX-C) surface and is able to induce synergistic biological activities including cell osteogenesis and proliferation.Multifunctional biomaterial surfaces can be created by controlling the competing adsorption of multiple proteins. To demonstrate this concept, bone morphogenetic protein 2 (BMP-2) and fibronectin were adsorbed to the hydrophobic surface of polychloro-para-xylylene. The resulting adsorption properties on the surface depended on the dimensional and steric characteristics of the selected protein molecule, the degree of denaturation of the adsorbed proteins, the associated adsorption of interphase water molecules within the protein layers, and the aggregation of proteins in a planar direction with respect to the adsorbent surface. Additionally, a defined surface composition was formed by the competing adsorption of multiple proteins, and this surface composition was directly linked to the composition of the protein mixture in the solution phase. Although the mechanism of this complex competing adsorption process is not fully understood, the adsorbed proteins were irreversibly adsorbed and were unaffected by the further adsorption of homologous or heterologous proteins. Moreover, synergistic biological activities, including cell osteogenesis and proliferation independently and specifically induced by BMP-2 or fibronectin, were observed on the modified surface, and these biological activities were positively correlated with the surface composition of the multiple adsorbed proteins. These results provide insights and important design parameters for prospective biomaterials and biointerfaces for (multi)functional modifications. The ability to control protein/interface properties will be beneficial for the processing of biomaterials for clinical applications and industrial products.Although the adsorption of molecules on material surfaces is a basic and intuitively understood phenomenon, the mechanisms governing protein adsorption are complex The present study employs a novel focus on utilizing functional proteins and exploiting the protein-surface interface to achieve a multifunctional platform for guided biological activities via the controlled adsorption of multiple functional proteins. The concentration of irreversibly bound protein molecules (compared to the fraction of reversibly desorbed proteins during the rinsing process) The PPX-C coating was prepared by a custom-built chemical vapor deposition (CVD) system comprising a sublimation zone, a pyrolysis furnace, and a deposition chamber. During the CVD polymerization process, dichloro-[2,2]-paracyclophane was first vaporized at approximately 150 °C and then transported to the pyrolysis furnace, where the dimer was pyrolyzed into monomer radicals at 670 °C. The radicals then entered the deposition chamber and polymerized on a rotating holder maintained at 15 °C to form a uniform PPX-C coating. To inhibit residual deposition, the temperature of the chamber wall was maintained at 90 °C. A stream of argon at a flow rate of 25 sccm was used as a carrier gas. Throughout the CVD polymerization, the operation pressure was regulated at 75 mTorr, and the deposition rate was maintained at approximately 0.5 Å/s, which was monitored by in situ QCM analysis (STM-100/MF, Sycon Instruments, USA).Recombinant human BMP-2 (R&D Systems, USA) and fibronectin from human plasma (R&D Systems, USA) were obtained commercially. BMP-2 was reconstituted as a stock solution at a concentration of 100 μg/mL in sterile 4 mM HCl and stored at −20 °C. Fibronectin was reconstituted as a stock solution at a concentration of 100 μg/mL in phosphate-buffered saline (PBS, pH 7.4, Sigma Aldrich, USA) and stored at 4 °C. Adsorption was performed by incubating the protein solutions on PPX-C-coated substrates at 4 °C for 10 min. After protein incubation, a rinsing protocol of three rinses with PBS (pH 7.4, containing Tween 20, Sigma-Aldrich, USA), one rinse with PBS (without Tween 20), and one rinse with deionized water was employed to remove the loosely adsorbed proteins. The mixed BMP-2/fibronectin protein solutions were prepared by varying the mixture ratio at 1:0, 10:1, 1:1, 1:10, and 0:1, based on mass concentration.Surface plasmon resonance (SPR) analysis was performed using a Biacore X-100 system (GE Healthcare, Sweden) equipped with a monochromatic and plane-polarized excitation light source. Standard SPR gold substrates (GE Healthcare, Sweden) were coated with PPX-C via the aforementioned CVD polymerization process, and a quartz prism was used to direct the light to the sample surface for analysis. A surface modified by PEG (containing thiol-terminal, MW 5000, Sigma-Aldrich, USA), which was immobilized on the maleimide-parylene-coated SPR substrate following a previously reported procedure The surfaces of cell culture plates (12 well, Corning, USA) were modified using the aforementioned PPX-C coating and protein adsorption procedures before use in cell culture. pADSCs isolated from subcutaneous adipose tissues pADSCs were seeded at a concentration of 1.5 × 104 cell/cm2 on the modified cell culture plates for 24 h. The cells were then incubated in standard osteogenic medium comprising DMEM/F12 supplemented with 10% FBS, 50 μM L-ascorbic acid (Sigma-Aldrich, USA), 100 nM dexamethasone (Sigma-Aldrich, USA), and 10 mM β-glycerophosphate (Sigma-Aldrich, USA). After 14 days of culture, ALP activity, alizarin red staining and gene expression were examined. For ALP analysis, cells were rinsed once with PBS (Sigma Aldrich, USA), fixed in 10% paraformaldehyde, and washed with distilled water. A staining procedure was then performed using an Alkaline Phosphatase Detection kit (Millipore, USA) in the dark for 20 min, followed by three washes with PBS (pH 7.4). For alizarin red staining, cells were incubated in 2% alizarin red staining solution (Sigma Aldrich, USA) for 20 min, followed by three washes with PBS (pH 7.4). The samples were then visualized using an Axio Scope.A1 microscope (Zeiss, Germany), and images were analyzed using the image analysis software ImageJ (NIH, USA) The proteins BMP-2 (M.W. 26 kDa) and fibronectin (M.W. 440 kDa) were selected based on their distinct structural dimensions, sizes, and diverse biological functions. Fibronectin is a key regulatory protein in processes such as cell adhesion, motility and proliferation; bone morphogenetic protein (BMP) is a growth factor involved in osteogenesis. A hydrophobic polymer coating of polychloro-para-xylylene (hereafter referred to as PPX-C), a highly biocompatible polymer (United States Pharmacopeia, USP, Class VI polymer) Sole adsorption of BMP-2 or fibronectin on the PPX-C surface was first characterized by surface plasmon resonance (SPR) to determine the mass of adsorbed protein based on the recorded optical signals a, 53.0 ± 3.2 ng/cm2 (2.0 ± 0.1 × 10−3 nmol/cm2) of BMP-2 and 261.5 ± 6.6 ng/cm2 (5.9 ± 0.2 × 10−4 nmol/cm2) of fibronectin were adsorbed on the PPX-C surfaces. The amount of adsorbed protein was inversely related to protein molecule size (26 kDa for BMP-2 and 440 kDa for fibronectin). Intermolecular steric hindrance and electrostatic repulsion The adsorption of BMP-2 or fibronectin on the surfaces was also characterized by quartz crystal microbalance (QCM). As shown in b, 101.0 ± 6.6 ng/cm2 (3.8 ± 0.2 × 10−3 nmol/cm2) of BMP-2 and 743.0 ± 65.0 ng/cm2 (1.7 ± 0.2 × 10−3 nmol/cm2) of fibronectin were adsorbed on the PPX-C surfaces; higher adsorption affinity was observed for both BMP-2 and fibronectin on PPX-C compared the PEG control surface. QCM analysis was further used to characterize the subsequent adsorption affinity for surfaces on which BMP-2 or fibronectin layers were already adsorbed. Homologous fibronectin (BMP-2 or fibronectin) or heterologous (BSA, 66 kDa; or FGF-2, 16 kDa) proteins were introduced to the first protein surface to dynamically investigate the binding affinity. As shown in c, low adsorption affinities ranging from 28.9 ± 2.7 ng/cm2 to 59.0 ± 2.6 ng/cm2 were detected for BMP-2, fibronectin, and BSA to adsorb on the first protein layers, comparable to the low affinities found on the PEG control surface (37.3 ± 3.3 ng/cm2, 18.0 ± 1.9 ng/cm2 and 30.1 ± 3.2 ng/cm2). Additional results of showing low adsorption affinity of a secondary FGF-2 protein to the first adsorption surfaces of BMP-2 and fibronectin were also included in the Supporting information. These results indicate that (i) the previously adsorbed BMP-2 or fibronectin saturated the adsorption capacity The adsorption results from both SPR and QCM were compared further based on the principles that the acoustic measurement of QCM includes the total amount of adsorbed molecules, and the SPR technique measures and converts optical signals to the mass of bound molecules of interest, and the results are independent of molecular morphology and of other adsorbed molecules d, a high level of interphase water (481.7 ± 60.0 ng/cm2) was observed for fibronectin. By contrast, a significantly lower degree of coupled interphase water (48.0 ± 3.4 ng/cm2) was observed for BMP-2. These results support the following conclusions. (i) The dry mass (excluding interphase water) of adsorbed fibronectin was 4.9 times greater than that of BMP-2, and based on the fixed mass/volume capacity for protein adsorption on an adsorbent surface, the occupied interphase thickness of the adsorbed proteins was proportional to the concentration of the dry mass Combined and competing adsorption of BMP-2 and fibronectin on the same PPX-C surface were further investigated. The binding affinity for a specific antibody was cross-examined by adsorbing varying ratios of BMP-2 and fibronectin, including 1:0, 10:1, 1:1, 1:10, and 0:1, on PPX-C surfaces. A combinatorial approach with cross-examination of the adsorbed protein mixture surfaces was subsequently performed by exposing the surfaces to human BMP-2 antibody and human fibronectin antibody. Binding was monitored by QCM (a and b). A high binding efficiency of the human BMP-2 antibody on the mixture surfaces was observed with increasing BMP-2 ratio, and the binding efficiency of the human fibronectin antibody similarly increased with increasing fibronectin content. The binding ratios of the two antibodies were calculated and normalized with respect to the combinatorial results, resulting in values of 1, 0.97, 0.51, 0.16 and 0 for BMP-2 and 1, 0.90, 0.57, 0.12 and 0 for fibronectin. These ratios were proportional to the individual solution concentrations of BMP-2 or fibronectin and were well correlated with the BMP-2/fibronectin mixture ratios of 1:0, 10:1, 1:1, 1:10, and 0:1. The kinetic analyses showed the binding constants were consistent with the theoretical values of 8.1 × 105 |
mL/μg and 5.1 × 105 |
mL/μg for BMP-2 and fibronectin to the antibodies, respectively c and imply the following. (i) The protein composition on a surface can be controlled by competing adsorption of multiple proteins. The resulting protein composition can be predicted and controlled by tuning the composition of the protein mixture in the solution phase, an important design parameter for the addition of functional proteins on the surfaces of biomaterials by adsorption. (ii) The correlation between concentration and binding was observed irrespective of protein size at the concentrations used (μg/mL range), which was consistent with the reported theory The topographical characteristics and the nanomechanical properties of the surfaces in the BMP-2/fibronectin system of competing adsorption on PPX-C surfaces were also investigated. Atomic force microscopy (AFM)-based nanoindentation utilizing the peak force tapping method a, the images from the indented modulus mapping revealed a significant boundary of granular structures on the surfaces of BMP-2/fibronectin adsorption, indicating tenaciously and irreversibly adsorbed BMP-2 or fibronectin on the PPX-C surface. A loosely bound protein surface should behave in a strand-like manner a. Young’s modulus exhibited distinct values with respect to small (0.63 GPa for BMP-2) and large granular (0.15 GPa for fibronectin) structures, further supporting the hypothesis of different protein aggregates. The quantitative results are shown in the bottom panels in a. However, images from the mapping of height (b) revealed relatively small variations of height throughout the entire surface, irrespective of the value of the BMP-2/fibronectin ratio. The root-mean-square roughness (Rq) was in the range of 0.7 ± 0.1 to 3.8 ± 0.2 nm. The interaction of hydrophilic groups with other molecules (e.g., neighboring protein or water molecules) has been attributed to an unfolded protein conformation Finally, the multifaceted and synergistic biological activities conferred by the adsorption of BMP-2 and fibronectin were examined by culturing porcine adipose stem cells (pADSCs) on the modified surfaces. Multifunctional properties were observed for both time latency and the functional specificity of the influence on the cells’ physiological activities, such as the cell growth/proliferation pattern and differentiation pathway. The cell growth and proliferation of pADSCs on the BMP-2 and fibronectin (varying ratios of 1:0, 10:1, 1:1, 1:10, and 0:1) surfaces were first examined after 24 h, and the MTT assay (a) revealed an increasing trend of cell viability. The number of pADSCs increased with fibronectin concentration, consistent with the enhancement of proliferation by fibronectin. Continued cell growth and proliferation were further verified by examining the cultured cells on the surfaces after 72 h. Microscopic observation revealed significantly confluent growth of the proliferated pADSCs with respect to cell shape and the cell number (b). The normalized ratio of the MTT analysis results at 72 h/24 h (c) increased with fibronectin concentration, further evidence of the regulation of the biological response by adsorption and the sustained effectiveness of the adsorbed fibronectin protein. Guidance of osteogenesis by BMP-2 was also investigated on the same surfaces. An early marker of osteogenesis, alkaline phosphatase (ALP), was analyzed after 7 days of culturing pADSCs on the modified surfaces to confirm the induction of osteogenesis by BMP-2, and a trend of enhanced osteogenic activity with increasing BMP-2 concentration was observed (a). The multifunctional protein surfaces were again examined after 14 days to detect the mature stage of osteogenesis. Alizarin red staining was first performed to examine the mineralization of the extracellular matrix, which is characteristic of osteogenesis in the mature stage b, a strong and positive trend of mineral formation with increasing concentration of BMP-2 on the modified surfaces was observed. The trend of enhancement of osteogenesis with increasing BMP-2 content was similar to that observed in the early-stage ALP analysis. In addition, direct evidence of guided osteogenesis was revealed by the up-regulation of osteogenic marker genes, including Runx2, osteocalcin, and osteonectin, in pADSCs after culturing (14 days) on the multifunctional protein surfaces. As summarized in c, trends of increased expression of these three genes with increasing BMP-2 concentration were observed, in contrast to the insignificant expressions of these genes on the surface with no BMP-2 (pure fibronectin) or on the pure TCPS and pure PPX-C control surfaces. The results for the expression of osteogenic genes were consistent with the results of alizarin red staining, and the osteoinduction induced specifically by BMP-2 was unambiguously verified on the same multifunctional surface. The osteogenic activities were also examined on various days of cell culture, and the results are included in the Supporting information. Taken together, the results above confirm that (i) the model system of multifunctional surfaces featuring adsorbed fibronectin and BMP-2 displays synergistic and concurrent multifunctions of cell proliferation and osteogenesis independently induced by fibronectin or BMP-2; and (ii) the defined surface composition of proteins and the resulting functional conduits exhibited specific and corresponding biological responses. A specific biological response can be controlled by the previously determined composition of the adsorbed functional proteins. Future biomaterials/biointerfaces should exploit these design parameters to produce multiple, tunable biological multifunctions (two or more than two) using a pre-determined mixture of functional proteins for adsorption on biomaterials/biointerfaces.Controlling protein molecules at the interface of biomaterials is challenging, and traditional engineering approaches, such as chemical conjugation, for the immobilization of protein molecules on surfaces are far from ideal The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript.The authors declare no competing financial interest.Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.colsurfb.2016.10.005Calculation of the relative interphase thickness of adsorbed proteins; Analysis of subsequent adsorption affinity by introducing secondary FGF-2 proteins to the first adsorption surfaces of BMP-2 or fibronectin; characterizations of osteogenesis activities.The following is Supplementary data to this article:Dynamic crack propagation simulation with scaled boundary polygon elements and automatic remeshing technique► A novel automatic dynamic crack propagation methodology was developed. ► Arbitrary n-sided polygons discretises the computational domain leading to flexible mesh generation. ► Generalised dynamic stress intensity factors determine the crack growth direction. ► A remeshing algorithm applicable to any polygon mesh accommodates crack propagation. ► Four dynamic crack propagation benchmarks were successfully modelled.An efficient methodology for automatic dynamic crack propagation simulations using polygon elements is developed in this study. The polygon mesh is automatically generated from a Delaunay triangulated mesh. The formulation of an arbitrary n-sided polygon element is based on the scaled boundary finite element method (SBFEM). All kind of singular stress fields can be described by the matrix power function solution of a cracked polygon. Generalised dynamic stress intensity factors are evaluated using standard finite element stress recovery procedures. This technique does not require local mesh refinement around the crack tip, special purpose elements or nodal enrichment functions. An automatic local remeshing algorithm that can be applied to any polygon mesh is developed in this study to accommodate crack propagation. Each remeshing operation involves only a small patch of polygons around the crack tip, resulting in only minimal change to the global mesh structure. The increase of the number of degrees-of-freedom caused by crack propagation is moderate. The method is validated using four dynamic crack propagation benchmarks. The predicted dynamic fracture parameters show good agreement with experiment observations and numerical simulations reported in the literature.vector of equilibrium generalised stress intensity factorsvector of generalised dynamic stress intensity factorsdynamic stiffness matrix of bounded mediumsingular stress modes at characteristic length, LDynamic fracture studies involve not only the singular stress field at the crack tip but also its interaction with stress waves that are reflected from the structure’s boundaries Today, the standard procedure for dynamic crack propagation simulations with FEM employs automatic local remeshing algorithms together with a rosette of singular quarter-point elements Many dynamic crack propagation problems have been modelled with the XFEM In the BEM, the geometry of the problem is vastly simplified because only the boundaries of the structure need to be discretised. Remeshing during crack propagation is also very simple because only new elements need to be added to the crack path as the crack propagates. This appealing feature is however, offset by the complex BEM formulations that involve analytical time and spatial integration of temporal shape functions In the SDG finite element method, the governing equations of equilibrium are discretised simultaneously both in space and time. This results in a system of equations that satisfies the balance of linear and angular momentum over every space–time element in the computational mesh to within machine precision. The formulation does not require any stabilization as it is dissipative. These properties allow the method to accurately resolve high stress gradients in the computational mesh caused by crack-tip fields and sharp wave fronts. The SGD finite element method was developed for linear elasto-dynamic fracture by Abedi et al. Recently, a number of numerical methods based on arbitrary n-sided polygons have been developed for structural analysis e.g. polytope elements with barycentric coordinates shape functions The scaled boundary finite element method (SBFEM) This study extends the polygon SBFEM to elasto-dynamic crack propagation simulations. A new local remeshing algorithm is developed to model rapidly propagating cracks that depend on the crack velocity. Compared to the remeshing algorithm developed in summarises the polygon SBFEM formulation for elasto-dynamics and the procedures used to extract the generalised dynamic SIFs. Section describes the development of the remeshing algorithm. Section demonstrates the application of the method to four dynamic crack propagation benchmarks. Section summarises the major conclusions that can be drawn from this study.Any domain, Ω can be discretised with a mesh of arbitrary n-sided polygons (a). Such a polygon mesh can be generated from a Delaunay triangulated mesh following the procedures outlined in b shows a polygon modelled by the SBFEM. Each edge on a polygon is discretised using one-dimensional line elements with local coordinates −1 ⩽ |
η |
⩽ 1. These elements can be of any order and does not affect pre-processing during mesh generation. Only linear elements are used in this study. A scaling centre is defined at the geometric centre of each polygon. The radial direction from the scaling centre to the boundary is described by a dimensionless radial coordinate, ξ, with ξ |
= 0 at the scaling centre and ξ |
= 1 at the boundary. The Cartesian coordinates with the origin at the scaling centre, (x, |
y) in a polygon are related to (ξ, |
η) by scaled boundary transformation equations where N(η) is the shape function matrix of line elements, and x0 and y0 are the vectors of polygon nodal coordinates.The governing equations for SBFEM in elasto-dynamics are now discussed. For brevity, only key equations will be presented. A detailed discussion can be referred to in where u(ξ) is the displacement in the radial direction and is obtained from the solution of the scaled boundary finite element equation in displacement for elasto-dynamicsE0ξ2u(ξ),ξξ+(E0-E1+E1T)ξu(ξ),ξ-E2u(ξ)+ω2M0ξ2u(ξ)+F(ξ)=0which describes the equilibrium in the ξ-direction , ω is the exitation frequency, F(ξ) is the load vector due to body loads and side-face tractions in the radial direction and the coefficient matrices E0, E1, E2 and M0 depend only on the polygon’s boundary and material propertieswhere D is the material constitutive matrix, ρ is the material density and B1(η) and B2(η) are strain–displacement matricesB1(η)=1|J|N(η),ηy000-N(η),ηx0-N(η),ηx0N(η),ηy0N(η)B2(η)=1|J|-N(η)y000N(η)x0N(η)x0-N(η)y0N(η),η|J|=(N(η)x0)(N(η),ηy0)-(N(η)y0)(N(η),ηx0)For the case of vanishing body loads and side-face loads, F(ξ) = |
0 in elasto-statics, ω |
= 0, Eq. can be transformed into a first-order ordinary differential equation according to ξu(ξ)q(ξ),ξ=-Zu(ξ)q(ξ)=-E0-1E1T-E0-1E1E0-1E1T-E2-E1E0-1u(ξ)q(ξ)where Z is a Hamiltonian matrix and q(ξ) is the internal nodal force vector can be decoupled into pairs of positive and negative eigenvalues using the block-diagonal Schur decomposition u(ξ)q(ξ)=Ψξ-Sc=Ψu(n)Ψu(p)Ψq(n)Ψq(p)ξ-S(n)S(p)c(n)c(p)where S and Ψ, are the eigenvalues and corresponding eigenvectors resulting from the Schur decomposition of Z. The matrix S is in Schur form. c is a vector of integration constants and ξ−S |
= |
e−Slnξ is a matrix exponential function The velocity and acceleration fields, u˙(ξ,η) and u¨(ξ,η) are simply the first and second time derivatives of Eq. The integration constants, c(n),c˙(n) and c¨(n) can be evaluated from Eqs. at the nodes on the polygon, η |
= −1 or η |
= 1 and ξ |
= 1 aswhere u0,u˙0 and u¨0 are the vectors of nodal displacements, velocities and accelerations on the polygon boundary. The static stiffness matrix, K is The mass matrix can be computed from the dynamic stiffness matrix, Sb(ω). Considering the case where there are no body loads and side-face loads, the amplitude of displacements u(ξ) are related to the internal nodal forces q(ξ) in the frequency domain by and differentiating with respect to ξ yieldsSb(ω,ξ),ξu(ξ)+Sb(ω,ξ)u(ξ),ξ-E0ξu(ξ),ξξ-(E0-E1T)u(ξ)=0 multiplied by ξ and simplifying, we obtainSb(ω,ξ),ξu(ξ)+(Sb(ω,ξ)-E1)ξu(ξ),ξ-E2u(ξ)+ω2M0ξ2u(ξ)=0 for ξu(ξ),ξ and substituting the result into Eq. (Sb(ω,ξ)-E1)E0-1(Sb(ω,ξ)-E1T)u(ξ)+Sb(ω,ξ),ξu(ξ)-E2u(ξ)+ω2M0ξ2u(ξ)=0 is valid for any arbitrary u(ξ) and hence, the coefficient matrices must vanish, leading toSb(ω,ξ)-E1E0-1Sb(ω,ξ)-E1T+Sb(ω,ξ),ξ-E2+ω2M0ξ2=0The dynamic stiffness matrix, Sb(ω) is a function of the dimensionless frequencyand the derivative with respect to af can be interpreted either as varying with ξ for a fixed ω or as varying with ω for a fixed ξ(Sb(ω)-E1)E0-1Sb(ω)-E1T+ωSb(ω),ω-E2+ω2M0=0In practice, Sb(ω) is usually not calculated. Instead, the low-frequency expansion of Sb(ω)neglecting higher order terms, O(ω4) is used in practice. M is the mass matrix of the polygon. For ω |
= 0, substituting Eq. , simplifying and neglecting higher order terms, O(ω4) yieldsω2ME0-1K-E1T+I+K-E1TE0-1+IM-M0+K-E1E0-1K-E1T-E2=0where, I is an identity matrix. The constant terms in Eq. . The terms in ω2 leads to the Lyapunov equation yields the mass matrix, M of the polygon.Using the stiffness and mass matrices, K and M, the conditions for equilibrium in a polygon can therefore be expressed aswhere F is the vector of nodal forces acting on the polygon. Eq. can be assembled polygon by polygon as in FEM and solved in the time domain using standard time integration algorithms The generalised dynamic SIFs, Kdyn(θ)=KI(θ)KII(θ)T for a crack propagating at velocity, c can be expressed aswhere kI and kII are the crack velocity functions and cd, cs and cR are the dilational-, shear- and Rayleigh wave speeds of the material, respectively.The equilibrium generalised SIFs, Keq(θ)=KIeq(θ)KIIeq(θ)T can be computed from stress field considering only the singular terms in S(n) as follows: The stress fields at any point in a polygon, σ(ξ, |
η) can be derived from Eq. and using the stress–strain relationship, σ(ξ, |
η) = |
Dϵ(ξ, |
η), where ϵ(ξ, |
η) is the strain field as), the real eigenvalues in S(n) will naturally include values between -1⩽Sii(n)⩽0. These eigenvalues describe the stress singularities at crack tips. Denoting these diagonal blocks in S(n) as S(s) where the superscript (s) stands for the diagonal blocks of S(n) with real eigenvalues -1⩽Sii(n)⩽0, with corresponding displacement modes Ψu(s) and integration constants c(s), the stress field considering only the singular modes, σ(s)(ξ, |
η) can be written in polar coordinates σ(s)(rˆ,θ) following a linear transformation where L is a characteristic length that is introduced so that the generalised equilibrium SIFs are independent of the system of units used that is computed using S(s),Ψu(s) and c(s). Ψσ(s)(η(θ)) is evaluated using Eq. with superscript ‘(s)’ replacing ‘(n)’ in S(n) and Ψu(n). Keq(θ) is defined by only the two stress components, σθθ(s)(rˆ,θ) and σrθ(s)(rˆ,θ)Defining a vector σ(s)(rˆ,θ)=σθθ(s)(rˆ,θ)σrθ(s)(rˆ,θ)T so that ΨσL(s)(θ) becomes a 2 × 2 matrix containing only entries corresponding to σθθ(s)(rˆ,θ) and σrθ(s)(rˆ,θ), the generalised equilibrium SIFs at angle θ is can then be used to compute the generalised dynamic SIFs.A crack is assumed to propagate in the direction of maximum circumferential stress and this can be determined from Kdyn(θ). The hypothesis used to determine the crack propagation direction is the angle θC that maximises KIdyn(θ) in Eq. Alternatively, the dynamic effects on crack propagation can also be modelled using a rate dependent damage evolution law. Guimard et al. The remeshing procedure for Case 1 is outlined in a shows a polygon mesh with a background triangular mesh (in green). This background triangular mesh is stored throughout the simulation for remeshing purposes. After Δa and θC has been determined from the crack velocity and Eq. , respectively, the location of the crack tip in the computational domain is first determined. Next, the boundaries of the triangular element in the background triangular mesh that is cut by the length Δa are determined. The boundary of the triangular element (A-B-C in b) is used to determine the boundaries of the set of triangular elements adjacent to it (J-K-L-M-N-C-O-P-Q). J-K-L-M-N-C-O-P-Q defines the area in which local remeshing is to be performed. If it is desired, a larger boundary can be constructed using the edges of triangular elements from a previous layer. This step, however, is usually not necessary. The triangular elements located inside J-K-L-M-N-C-O-P-Q are then removed from the background triangular mesh (A Delaunay triangulation is then constructed on the empty patch (d) with the constraint that only one triangular element is allowed on each edge of the boundary of the patch. This allows the newly constructed triangular elements to be compatible with the remainder of the background triangular mesh. In this study, the remeshing algorithm is linked to the commercial finite element package, ABAQUS The remeshing procedure for Case 2 is outlined in for one crack propagation step. It resembles the procedure in Case 1 outlined in . The only difference is in detecting the boundaries of the triangular elements that are cut by the length Δa. Here, the boundary that is used to determine the set of triangular elements adjacent to it comprises of all the triangular elements that are cut by Δa. In a, for example, Δa cuts across two triangular elements in the background mesh. The boundary A-B-C-D that is made up from triangular elements ABC and ACD determines the area J-K-L-M-N-D-O-P-Q-R in which local remeshing is to be performed. that this local remeshing procedure only minimally alters the global mesh structure in both cases. Compared to the procedure previously devised in After remeshing, mesh mapping is necessary to transfer the displacement, velocity and acceleration fields from the old mesh to the new mesh to be used as initial conditions in the following time step. The displacements, velocities and accelerations are computed directly from Eqs. . Specifically, for a point located at Cartesian coordinates (xP, |
yP) in the new mesh after remeshing, the polygon in the old mesh within which the point (xP, |
yP) is located is first found. These are then transformed into SBFEM coordinates in the old polygon (ξP, |
ηP) using scaled boundary transformation equations Four dynamic crack propagation benchmarks are modelled using the polygon SBFEM. In all the problems presented in this section, each polygon edge is discretised using one linear finite element. Each edge on a polygon containing a crack is discretised with five linear elements so that accurate Kdyn(θ) can be computed. In dynamic crack propagation simulations, the crack propagation length is proportional on the instantaneous crack velocity and the size of the user specified time step, Δt. For small values of Δt, the crack propagation length accordingly decreases. Although it is possible to perform the analysis as it is, such a small resolution in the crack propagation length is not necessary. This is because the frequency in which the remeshing algorithm is called increases the mesh density but does not result in significant increase in solution accuracy. An approximate approach is adopted in this study by introducing a resolution tolerance length, Δatol. During each time step, the instantaneous crack propagation length is accumulated until its magnitude is greater than Δatol before propagating the crack. The fracture parameters i.e. Kdyn(θ) and crack velocity are then averaged throughout the duration in which the crack propagation length has been accumulated. This approach results in significant savings in the number of DOF required for a dynamic crack propagation simulation without over sacrificing the solution accuracy. All computations were performed on an Intel (R) Xeon (R) @3.33 GHz CPU desktop.a shows a rectangular double cantilever beam (RDCB) held by two pins that was experimentally tested by Kalthoff et al. b shows the polygon mesh of the RDCB. The mesh consists of 465 polygons and 1128 nodes.Prior to dynamic crack propagation simulation, a static analysis was performed by applying vertical displacements at both pins until the mode-I SIF, KI=2.32MPam corresponding to the initial SIF reported in the experiments The fracture parameters predicted by the polygon SBFEM were compared with the experimental measurements shows the predicted KIdyn versus crack length, a relation. shows the predicted crack velocity history. As is evident from the figures, all the predicted fracture parameters agreed well with the experimental data and other numerical methods reported in the literature. The predicted KIdyn showed a similar trend as the FEM predictions of Nishioka et al. shows the final polygon mesh. Compared with the initial mesh, the number of polygons and nodes had increased to 705 and 1754, respectively. The increased DOFs in the mesh is, however, still small compared with that usually required in a FEM analysis.A notched Polymethlyl Methacrylate (PMMA) beam subjected to an impact load tested by Nishioka et al. a is considered. The impact velocity is 5 ms−1. The material properties of PMMA are: Young’s modulus, E |
= 2940 MPa, Poisson’s ratio, v |
= 0.3, density, ρ |
= 1190 kg m−3, Rayleigh wave speed, cR |
= 1064 ms−1, shear wave speed, cs |
= 941 ms−1 and dilatational wave speed cd |
= 1710 ms−1. The thickness of the beam is 10 mm and l |
= 20 mm. Plane stress conditions were assumed. The polygon mesh used to discretise the beam is shown in b. It has 342 polygons and 838 nodes. A mixed-phase type-A dynamic fracture simulation as classified by Nishioka et al. shows that the dynamic SIF histories predicted by the polygon SBFEM compared very well with the experimental measurements and FEM simulations of Nishioka et al. shows a close-up of the predicted crack propagation process. The crack propagation path was initially curved, emanating from the crack tip and propagating at an angle of approximately 130°. It then gradually turned towards the point of impact. The final polygon mesh at t |
= 212 μs shown in e shows the final configuration of the beam. Although the number of DOFs had increased as a result of remeshing, the total number of DOF is still much smaller compared to that required by the FEM. shows that the crack path predicted by the present method compared very well with the FEM simulations in a) subjected to impact loads at both ends is considered. Experiments on such specimens were reported by Grégoire et al. Time dependent loads were applied to both ends of the specimen according to the BEM simulation of Fedelinski b. The resolution tolerance length used to determine crack propagation was Δatol |
= 1.7 mm. The total computational time used to solve this example was 596.639s i.e. approximately 10 minutes. compares the dynamic SIF histories predicted by the polygon SBFEM with those of the BEM simulations of Fedelinski shows the predicted crack propagation process. The crack initially propagates at an angle of approximately 340°until t |
= 270 μs. Between 320 μs ⩽ |
t |
⩽ 500 μs, the crack first reoriented itself until it was almost parallel with the horizontal axis. Then it propagated in almost a straight line. The final polygon mesh shown in compares the crack paths predicted by the polygon SBFEM and BEM simulations. Both crack paths compared very well with each other.A 18Ni1900 steel plate with two parallel edge notches subjected to an impact load between the two notches, as shown in a, is considered. The impact velocity is V |
= 16.5 ms−1. The boundaries of the plate are traction free. The material properties of the plate are: Young’s modulus, E |
= 200 GPa, Poisson’s ratio, v |
= 0.3, density, ρ |
= 8000 kg m−3 and static critical stress intensity factor, KIC=68MPam. Plane strain conditions were assumed. Experimental investigations on such specimens were reported by Kalthoff and Winkler b shows the polygon mesh that is used to discretise the plate. It has 305 polygons and 758 nodes. Due to the symmetry of the problem, only half the plate is modelled. A time step of Δt |
= 1 μs was used for the simulation. The resolution tolerance length used to determine crack propagation was Δatol |
= 1 mm. The total computational time used to solve this example was 774.619s i.e. approximately 12 minutes. compares dynamic SIF histories predicted by the polygon SBFEM with the EFG and FEM-EFG simulations of Belytschko et al. show the polygon SBFEM predicted crack propagation process. Crack propagation was almost a straight line at an angle of approximately 74°, measured from the crack tip. This agreed very well with the observations made by Kalthoff and Winkler d, the number of polygons and nodes in the mesh had increased to 483 and 1171, respectively. The increase in number of DOFs was still small compared with the FEM-EFG simulations in shows that the crack path predicted from the current study agreed well with the EFG simulations in Dynamic crack propagation problems have been successfully simulated using the polygon SBFEM. Arbitrary n-sided polygons were used to discretise the computational domain. Each polygon in the mesh was treated as a SBFEM subdomain. Standard SBFEM procedures were used to compute the stiffness matrix, load vectors in each polygon and the generalised dynamic SIFs in a cracked polygon. Crack propagation was treated using an automatic local remeshing algorithm that can be linked to standalone triangular mesh generators or the pre-processor module of commercial FEM codes. The algorithm was successfully applied to dynamic fracture problems where the crack paths depended on the instantaneous crack velocity. The developed method was successfully validated using four dynamic crack propagation benchmarks. The crack propagation paths and dynamic fracture parameters that were predicted by the polygon SBFEM agreed well with the physical cracking phenomena that are reported in experiments and other numerical results in the literature. It was found that the developed method has the following attractive features:Stress singularities of any kind that are described in the form of generalised dynamic stress intensity factors can be accurately computed from the matrix power function solutions of singular stress fields in the SBFEM following standard finite element stress recovery procedures.The area in the vicinity of the crack tip does not need to be discretised with a very fine mesh as in FEM. There is also no need for any enrichment functions, special treatment at the crack tip and integration rules such as in XFEM or some meshless methods.The use of polygons allows complex geometries to be discretised with more flexibility compared to 3-noded triangular- and 4-noded quadrilateral finite elements.The local remeshing algorithm that was developed only minimally changes the mesh topology throughout the entire simulation. This increases the accuracy of mesh mapping because there are only few nodes in which the state variables needs to be mapped to the new mesh.It was observed that the polygon SBFEM required significantly less number of DOF compared to other numerical methods reported in the literature e.g. FEM and meshless methods.The developed method is thus a competitive alternative to FEM- and XFEM- based approaches for dynamic crack propagation modelling. Recent developments on XFEM and meshless methods showcased their capability to solve multiple crack propagation in three dimensions. The extension of the polygon SBFEM to three-dimensional crack propagation modelling is certainly possible. An n-faceted polyhedra can be generated from a tetrahedral mesh following a similar approach of generating an n-sided polygon mesh from a mesh of triangular elements. Each polyhedra can then be modelled as a SBFEM subdomain in three dimensions. The modes that contribute to the singular stresses can then be extracted from an equivalent n-faceted crack polyhedra to evaluate the crack propagation criterion and direction. The concepts developed in the automatic remeshing algorithm discussed in this study can also be extended for three-dimensional crack propagation modelling wherein the background triangular mesh used to generate an n-sided polygon mesh is replaced by a background tetrahedral mesh that is used to generate an n-faceted polyhedra mesh. The extension of the polygon SBFEM to three-dimensional crack propagation modelling would be an interesting research in the near future.Fire resistance of ultra-high performance strain hardening cementitious composite: Residual mechanical properties and spalling resistanceUltra high performance strain hardening cementitious composites (UHP-SHCC) is a special type of cement-based composite material with outstanding mechanical and protective performance at room temperature. But its fire performance is unknown and there is a lack of research in this aspect. This study presents an experimental program to study fire resistance of UHP-SHCC under two aspects, viz. high-temperature explosive spalling resistance and residual mechanical performance after a fire. Both compressive strength and tensile strength of UHP-SHCC were found to deteriorate with increasing exposure temperature. Tensile strain-hardening feature of UHP-SHCC would be lost at 200 °C and above. It was found that PE fibers are found not effective in mitigating explosive spalling, although they start to melt at 144 °C. FE-SEM (Field Emission Scanning Electron Microscopy) and EDX (Energy Dispersive X-ray) techniques were used to study the state of fiber, fiber/matrix interaction, and microcracks development. Microscopic study found that melted PE fibers were still present in the cementitious matrix, and the melting did not introduce more microcracks. Furthermore, it was difficult for melted PE fibers to diffuse through the matrix, thus providing the reason that PE fibers did not mitigate explosive spalling in UHP-SHCC.There is a growing trend in tailoring concrete for high tensile ductility in the past few decades. This type of high ductility concrete is known as strain hardening cementitious composite (SHCC) [], or engineered cementitious composite (ECC) [] if designed based on the principles of micromechanics. SHCC is distinct from concrete and fiber reinforced concrete (FRC) in terms of deformation behavior under uniaxial tension. As illustrated in , in contrast to brittle behavior of concrete ((a)) and strain softening behavior of FRC ((b)), SHCC exhibits pseudo strain-hardening behavior due to development of multiple fine cracks ((c)). Tensile strain capacity of SHCC ranges from 1% to 5%, 100–500 times that of ordinary concrete. However, compressive strength of SHCC ranges from 20 to 60 MPa [In the past decade, researchers have developed SHCC with high mechanical strength and protective performance, termed as UHP-SHCC []. To engineer SHCC for ultra high performance, a high strength cement-based matrix should be used. However, polyvinyl alcohol (PVA) fibers commonly used in SHCC are not eligible for developing UHP-SHCC. This is because relatively low tensile strength and hydrophilic nature of PVA fibers cannot meet the requirements for steady-state and multiple cracking []. If PVA fibers were to be adopted in high-strength matrix, they would fracture other than pull out upon occurrence of a crack. Therefore, instead of PVA fibers, PE fibers with higher tensile strength and greater elastic modulus, are commonly used to develop UHP-SHCC []. In contrast with hydrophilic nature of PVA fiber, PE fiber is hydrophobic. These advantages make PE fibers suitable for developing UHP-SHCC.] contributed to understanding of fire resistance of SHCC. Compared to SHCC, fire performance of UHP-SHCC is much less understood. To date, there is no published work on fire resistance of UHP-SHCC.In fact, fire resistance of UHP-SHCC is questionable. Melting point of PE fibers is 144 °C, even lower than melting point of PVA fibers (240 °C), as shown in DSC curves of the two fibers in . Loss of PE fibers would have two potential impacts on fire resistance of UHP-SHCC. On one hand, PE fiber is an indispensable element in achieving pseudo tensile ductility. Hence, the loss of PE fibers means distinct feature of ductility vanishes from UHP-SHCC. On the other hand, melting of PE fibers tends to influence moisture migration inside UHP-SHCC, consequently reducing spalling risks. Spalling, an unfavorable phenomenon frequently occurring in fire testing of concrete, may take place if pore pressure due to trapped water vapor exceeds the tensile strength of concrete. It reduces concrete section and weakens load-bearing capacity of concrete member at the early stage of a fire. Previously, no specific test has been conducted to examine explosive spalling tendency of UHP-SHCC. Whether the influence of PE fibers is positive or negative in mitigating spalling and how PE fibers function to exert the influence remain unknown.] added only PE fibers in UHP-SHCC. Recent study found that steel fibers improved the strength and ductility of SHCC after exposure to elevated temperatures []. Besides, fiber cocktails are more effective to mitigate explosive spalling than single type of fibers []. Therefore, hybrid PE/steel fibers, instead of PE fibers, were used in the UHP-SHCC mix studied in this paper.The objective of this research is to study fire performance of UHP-SHCC with hybrid PE/steel fibers and the mode of action of PE fibers in mitigating explosive spalling. Fire performance of UHP-SHCC is examined in two aspects in this paper: explosive spalling resistance of UHP-SHCC at high temperature and residual mechanical properties of UHP-SHCC after exposure to elevated temperatures. To study the effect of PE fibers on explosive spalling resistance, mortar specimens (without any fiber) were used as control samples. FE-SEM and EDX techniques were used to study the states of fibers, microcracks development, and fiber/matrix interaction after moderate heat treatment to explore possible means of PE fibers to combat explosive spalling.The mix proportions of UHP-SHCC are given in . In this mixture, CEM I 52.5N was used, and the content of silica fume was 25% of the content of cement. High strength PE fiber and steel fiber were chosen for UHP-SHCC. The properties of PE and steel fibers are listed in To produce UHP-SHCC, cement and silica fume were first mixed for approximately 5 min at dry state. After that, water and superplasticizer were added into the solid mixture and mixed until a good workability was achieved. Finally, PE and steel fibers were added sequentially to the fresh matrix and the mixture was stirred for 5 more mins to ensure no clumps of fibers could be felt in the matrix. Fresh UHP-SHCC was then cast into molds and compacted using a vibration table. After casting, a plastic sheet was placed on top of the molds to prevent moisture loss. The specimens were demolded after 1 day and cured in water for 27 days. After that, the specimens were conditioned in ambient lab environment until the day of testing.To determine residual mechanical properties of UHP-SHCC, cylindrical specimens (ϕ100 × 200 mm) and dog-bone specimens () were used for uniaxial compressive and tensile tests, respectively. The setups for compressive and tensile tests are shown in (a) and (b), respectively. A displacement-controlled loading regime was chosen for both compressive and tensile tests. The displacement loading rate was 0.2 mm/min.To capture the full compressive stress-strain curve, compressive force was measured by a load cell and axial displacements recorded by one set of LVDTs ((a)). Three LVDTs were attached on the center region of the cylindrical specimen forming an angle of 120° between consecutive LVDTs ((c) to ensure more accurate measurements of average compressive strain. The gage length of the three LVDTs was 100 mm. However, the readings of these three LVDTs were only valid up to the peak load level. After attaining the peak load, cracks in the specimen further widened rendering the three LVDT readings rather erratic. Therefore, two additional LVDTs were used to measure deformations of the specimen between the machine platens. The deformations included the compressive strain of the specimen, the end-zone effect and the movements of the machine platens. A correction factor proposed by Mansur et al. [] was adopted to obtain the post-peak branch of compressive strain-strain curve of UHP-SHCC.To develop tensile stress-strain curves of UHP-SHCC, tensile force was recorded by a load cell and displacements measured by two external LVDTs attached to the dog-bone specimen with a gage length of 100 mm as shown in (b). For the compressive tests, the cylindrical specimens were subjected to the following isothermal temperatures: 30 °C (ambient temperature), 200 °C, 400 °C, 600 °C, and 800 °C. For the tensile tests, the dog-bone specimens were subjected to the following temperatures: 30 °C (ambient temperature), 100 °C, 200 °C, 300 °C, 400 °C, 500 °C, and 600 °C. The temperature interval for the tensile tests is set to be smaller than that for the compressive tests, since tensile properties of UHP-SHCC are more sensitive to temperature change.Prior to heating, the compressive and tensile specimens were dried at 105 °C to constant mass if the target exposure temperature was higher than 105 °C. This was to remove the influence of excessive water content on the mechanical properties of UHP-SHCC as well as to avoid explosive spalling. After that, the specimens were heated to target temperature at 1 °C/min. A low heating rate was adopted to prevent explosive spalling during heating phase, since spalling risk is high when heating rate is large []. After reaching target temperature, the specimens were immersed in that environment for 1 h to achieve isothermal condition before cooling down naturally in the furnace. gives the total number of specimens for compressive and tensile tests. For each temperature exposure level, three specimens were prepared. All the specimens were tested after 28 days.To evaluate spalling resistance of UHP-SHCC, cylindrical specimens (ϕ100 × 200 mm) were prepared. To demonstrate the effect of PE fibers in combating explosive spalling, high strength mortar specimens were prepared with the same mix design but without any fibers as given in . To exclude the influence of steel fibers on the spalling resistance, specimens made of PE-FRCC were also prepared so that only the effect of PE fibers was studied. The mix proportions of PE-FRCC are presented in gives the total number of specimens for spalling tests.The cylindrical specimens were cured in water for 28 days. They were then conditioned in ambient lab environment until the age of 90 days to achieve a balanced moisture state with ambient environment within the specimens. Then the specimens were heated to 400 °C in 7 min in the furnace, and the temperature was held constant for 2 h. A high heating rate of 53 °C/min was adopted to study spalling sensitivity, since it is more closer to real fire scenarios and more likely to trigger explosive spalling []. A perforated steel cage was used to cover the specimens to prevent spalling debris from damaging the internal heating elements while allowing heat convection to occur.To study how PE fibers function to combat explosive spalling, FE-SEM and EDX techniques were used to study microstructural changes of UHP-SHCC and fiber status after exposed to 30 °C, 105 °C, 150 °C, and 200 °C. The UHP-SHCC sample for FE-SEM and EDX tests was a small cylinder (ϕ 12 × 12 mm) and was prepared following the same way as the UHP-SHCC specimens. The sample was heated to the target temperature at 1 °C/min and held in the isothermal condition for 15 min. After that, the sample was allowed to cool down at 1 °C/min to minimize micro-cracks due to thermal gradients in the sample. shows the UHP-SHCC specimens after heated to 200 °C, 400 °C, 600 °C, and 800 °C, respectively. No micro-cracks were visible after heated to 200 °C as shown in (a). But after being heated to 400 °C, micro-cracks were visible on the specimen surface ((b)), and the crack width and density of cracks increased as exposure temperature increased to 600 °C ((d)). This is due to shrinkage of UHP-SHCC at high temperature as evidenced by change in length of UHP-SHCC as shown in . Thermal contraction of UHP-SHCC was measured using a DIL 802 differential dilatometer. The heating rate was set to be 1 °C/min, the same as that used for residual mechanical tests. As shown in , UHP-SHCC started shrinking at 271 °C, and the shrinkage increased with temperature rise. The increase in shrinkage became more rapid at 700 °C and above. This explains the micro-cracks observed in UHP-SHCC specimens after exposure to 400 °C and above.Compressive stress-strain curves of specimens at ambient temperature and post-heated specimens are presented in and typical failure patterns of specimens after exposure to elevated temperatures are shown in . The compressive behaviors of UHP-SHCC at ambient and after heated to 200 °C are quite similar as seen in (a) and (b). Their failure patterns were characterized by a major inclined shear crack forming along the height of the specimens. For specimens subjected to 400 °C and 600 °C, the pre-peak stress-strain curves were fairly nonlinear as shown in (c) and (d). The initial slopes of their compressive stress-strain responses were lower than the slopes in the middle stage of tests, indicating that the initial cracks in UHP-SHCC closed up during compressive loading. At the beginning of loading the cracks were closing as the load gradually increased, resulting in a lower initial slope. After the cracks were closed, the slopes of the curves gradually increased and followed the general trend of a compression test. Multiple interacting cracks were observed on the specimens after heated to 400 °C and 600 °C as shown in (c) and (d). For specimens subjected to 800 °C, the compressive behavior ((e)) was very different from all the others mentioned above ((a)–(d)). This could be caused by large thermal crack density and crack width since the specimens were unloaded. As shown in (e), the UHP-SHCC specimen was completely crushed at the middle part. gives compressive strength of UHP-SHCC and ordinary SHCC as a function of isothermal temperature. Each data point in represents the mean value of three samples. Mix proportions of SHCC are given in . The same compressive test setup and loading scheme as used for UHP-SHCC were adopted for SHCC. Cylinder specimens of the same dimensions as those of UHP-SHCC specimens were used for residual compressive tests on SHCC. For SHCC, the heating rate was 10 °C/min and the dwelling time after reaching target temperature was 2 h. More information about residual compressive properties of fire-damaged SHCC can be found in Ref. []. Compressive strength of UHP-SHCC at room temperature is 117.6 MPa as shown in . After exposure to 200 °C, it decreases slightly to 113.1 MPa. However, as exposure temperature increases to 400 °C and 600 °C, compressive strength of UHP-SHCC drops to 65.6 MPa and 56.9 MPa, respectively. When the exposure temperature reaches 800 °C, only 12.3% of the compressive strength of unheated specimens (14.5 MPa) remains.Compressive strength of UHP-SHCC is much larger than that of SHCC at 30 °C and 200 °C as shown in . But after exposure to 400 °C and 600 °C, compressive strength values of UHP-SHCC are only slightly larger than those of SHCC. After 800 °C, compressive strength of SHCC surpasses that of UHP-SHCC.From the perspective of strength reduction percentage, compressive strength of UHP-SHCC deteriorates much faster with elevated temperature than that of SHCC as presented in . Microstructure pore coarsening and decomposition of CH and CSH gel are two major factors for strength degradation of SHCC after exposure to elevated temperature []. For UHP-SHCC, in addition to those two factors, the shrinkage-induced cracks as shown in (b)–(d) contribute to strength deterioration significantly. shows a SHCC specimen after exposure to 800 °C. It is quite different from the UHP-SHCC specimen after exposure to 800 °C, as no obvious cracks are visible on the surface of the SHCC specimen by naked eyes. This explains the different trends in compressive strength reduction between SHCC and UHP-SHCC after exposure to 400 °C and above. Residual compressive performance of UHP-SHCC is even poorer than those of siliceous and calcareous concrete [Tensile stress-strain curves of UHP-SHCC at 30 °C and after heated to 100,200,300, 400, 500, and 600 °C are presented in (a)–(g). In general, both tensile strength and strain capacity of UHP-SHCC decrease with an increase in exposure temperature, as shown in (a)–(g). UHP-SHCC has a tensile strength of 5.7 MPa and a tensile strain capacity of 2.2% at room temperature by averaging the tensile test results in (a). After exposure to 100 °C, tensile strength and strain capacity of UHP-SHCC decrease to 4.8 MPa and 1.8% by averaging the three tensile test results in (b). At 200 °C and above, tensile behaviors of UHP-SHCC ((c)–(g)) are quite different from those of UHP-SHCC at 30 °C and 100 °C ((a) and (b)). UHP-SHCC exhibits pseudo strain-hardening behavior at 30 °C and 100 °C but shows strain-softening behavior from 200 °C to 600 °C. This is because PE fibers start to melt at 144 °C as shown in The temperature-dependent tensile strength reduction factors for UHP-SHCC under various temperatures are plotted in ], tensile strength of UHP-SHCC reduces rapidly upon reaching 400 °C. Though 29.3% of tensile strength of UHP-SHCC remains after heated to 600 °C, it is a very small capacity and is negligible. show the UHP-SHCC and reference mortar specimens before and after heating, respectively. The UHP-SHCC specimens remained intact ((a)), but there was a large circumferential crack in one UHP-SHCC specimen. If not for the bridging action of steel fibers, the UHP-SHCC specimens would have broken into various pieces, such as the reference mortar specimens as shown in To confirm this finding another three specimens made of PE-FRCC were prepared for spalling tests. The mix proportions of PE-FRCC specimen are almost the same as those of UHP-SHCC, except that no steel fibers are used in PE-FRCC (). The testing method for PE-FRCC specimens followed that specified in Section shows the PE-FRCC specimens after heating. Severe spalling occurred as evidenced by many broken pieces. This proves that hybrid steel/PE fibers are more effective than mere PE fibers in mitigating explosive spalling at high temperature.Another interesting finding was that addition of 12.5 kg/m3 of PE fibers did not prevent explosive spalling, although they have a melting point as low as 144 °C. In contrast, PP fibers which melt at about 160–170 °C, can effectively prevent explosive spalling at a dosage of 2 kg/m3 []. Therefore, there is a need to investigate why PE fibers are not effective in prevent spalling.It is widely accepted that explosive spalling of concrete is mainly due to pore pressure buildup inside concrete at elevated temperature [] and PP fibers can mitigate spalling by increasing permeability of concrete []. But how PP fibers function to increase permeability of concrete is a controversial issue. Some researchers think melting of PP fibers generates connected empty spindly channels, thus increasing permeability of concrete []. Others think that microcracks induced by thermal expansion of PP fibers are the source of permeability increase [To check whether PE fibers create empty channels after melting, FE-SEM was used to characterize status of PE fibers and fiber/matrix interaction before and after melting of fibers. (a) and (b) show SEM images of PE fibers before and after heated to 200 °C, respectively. It can be seen that the melted PE fiber was still present inside UHP-SHCC at 200 °C. This finding was further confirmed by thermogravimetric (TG) curve of PE fibers as presented in . The TG analysis was performed with a heating rate of 10 °C/min and a nitrogen flow rate of 20 mL/min. As shown in the figure, PE fiber does not vaporize until about 500 °C. also plots the TG curve of PP fiber. It is interesting to find that PP fiber has a higher melting point than PE fiber but a lower vaporization point. It could be one reason that explains PE fibers are less effective than PP fiber in preventing explosive spalling.To find out the penetration behavior of melted PE fibers, elemental mapping was conducted for the area shown in shows the corresponding EDX curve and element composition. It further confirms that melted PE fibers are still present in the cementitious matrix. shows the distribution of carbon element in that area after exposure to 200 °C. A large portion of PE fiber residue is still present in the fiber channel, with a small portion of PE fibers diffusing into microcracks and cementitious matrix. The penetration depth of molten PE fiber into the matrix is very small, approximately 20 μm, due to its high viscosity. It could be another factor accounting for inability of PE fibers to suppress explosive spalling.To find out whether PE fibers initiate microcracks in the matrix around them, SEM images were taken at a location where no initial microcrack was present around a PE fiber before and after exposed to 200 °C as shown in (a) and (b). No microcracks were observed around the PE fiber as shown in the figure.To further study the potential influence of PE fibers on microcrack development, a consecutive series of SEM images (500× magnification) were taken on the same location of a UHP-SHCC sample at room temperature (30 °C) and after exposure to 105, 150, and 200 °C. For each exposure temperature, the images were stitched together and microcracks in the stitched image were identified by using adaptive threshold binarization. For comparison purpose, a reference mortar sample was also prepared for crack development analysis. show microcracks of mortar and UHP-SHCC sample after exposure to 30, 105, 150, and 200 °C, respectively. Obviously, both the crack lengths and widths in both the mortar and the UHP-SHCC samples increased with increase in temperature. To quantify microcrack development of mortar and UHP-SHCC, crack density of mortar and UHP-SHCC was calculated according to Eq. with A as the material surface area, n is number of cracks, and li is ith crack length. shows the crack density of mortar and UHP-SHCC as a function of exposure temperature. Crack density of both mortar and UHP-SHCC increased with temperature. But the increase rate in crack density of UHP-SHCC was much lower than that of mortar. It again proves that addition of PE fibers does not introduce additional microcracks.Fire resistance of UHP-SHCC was examined from two aspects in this paper, i.e., residual mechanical properties and thermal spalling behavior. UHP-SHCC was found to perform poorly at high temperature, although it has excellent performance at ambient condition. The detailed findings are summarized as follows:After exposure to 400 °C and above, severe cracking was observed on the surface of UHP-SHCC specimens. The severity of cracking increased with temperature. This was because UHP-SHCC shrank as temperature increased.Compressive strength of UHP-SHCC decreased as temperature increased. The decreasing trend was especially obvious when UHP-SHCC specimens were exposed to 400 °C and above. After heated to 800 °C, UHP-SHCC only retained 12.3% of its ambient compressive strength. In general, deterioration in compressive strength of UHP-SHCC was consistent with the surface crack development of UHP-SHCC specimens after elevated temperature.Similar to compressive strength of UHP-SHCC, its tensile strength also decreased with temperature increase. Tensile strain capacity of UHP-SHCC decreased as temperature increased from room temperature to 100 °C. From 200 °C to 600 °C, UHP-SHCC showed tensile strain-softening behavior instead of strain-hardening behavior.UHP-SHCC spalled violently after exposed to 400 °C and PE fibers were found not effective in mitigating explosive spalling of UHP-SHCC although they have a melting point of 144 °C. After melting, a majority of melted PE fibers were still present inside their channels and did not penetrate deep into the matrix. Furthermore, PE fibers did not introduce additional microcracks after melting. These two factors explain inability of PE fibers in preventing explosive spalling.Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the L2 NIC.The interaction between mass transfer effects and morphology in heterogeneous olefin polymerizationA new mathematical model describing the build-up and relaxation of elastic tensions inside growing particles during the heterogeneous polymerization of olefins is described. Mass transfer resistance can lead to spatial variation in the expansion rate of polymer which creates tension. This can cause the particle to break up. The reciprocal interaction between morphology and mass transfer resistance is examined by applying the model to two cases where experimental data is available. It was found that the model predicts the formation of hollow particles in the presence of severe mass transfer limitations, in accordance to what is found experimentally. Model results predict that rupturing occurs early in the course of polymerization when monomer concentration gradients are at a maximum inside the growing particles.Understanding the interaction between morphology, mass transfer and kinetics is vital in the design of new catalytic systems, and in the scale-up of laboratory processes. Consider what happens to a particle once it is injected into the reactor. Immediately upon injection, monomer starts to flow into the pores of the catalyst particle, and polymer forms on the surface of the active sites. Monomer continues to flow into the pores if they do not fill up, then through the layer of polymer. Once enough polymer forms, the support fragments and the micrograins move apart, but the original particle maintains its integrity due to the polymer already formed. The forces created by the formation of polymer cause the expansion of the particle, and the tension that is thus induced in the particle in a sense controls the particle growth. To the best of our knowledge, no attempts at modelling this aspect of particle growth (i.e. the rate of expansion and the morphology of the particle) have been published in the open literature. For this reason, we derived a new polymer particle growth model that incorporates the creation of tensions inside the particle due to uneven growth, and have examined the two-way interaction between morphology and mass transfer resistance in heterogeneous olefin polymerization.Ideally, a single particle model should describe the relation between reaction rate, growth of the particle, development of the particle morphology and evolution of polymer properties like average molecular weight (Mn,Mw), polydispersity (q=Mw/Mn), chemical composition distribution in (CCD) and crystallinity. This requires a kinetic model that describes reaction rate as function of local chemical composition and state variables (T,P), coupled to a model that describes how these variables are distributed in the particle, and a model describing the evolution of particle morphology. It is the latter which is focused here, keeping the kinetics as simple as possible.Most existing models for particle growth are based on variations of the well-established multigrain model (MGM), which has been extensively described in the literature, e.g. Floyd, Choi, Taylor, and Ray (1986). This model is based on experimental observations that show the original catalyst particle breaking up into fragments (micrograins) that are dispersed in the growing polymer particle (macroparticle). In the MGM, concentration gradients can be taken into account at both the macro- and microparticle scale. However, it has been shown by and others, that the microparticle gradients are generally negligible, and a polymer flow model (PFM) (), that assumes a homogeneous distribution of catalyst in the polymer phase, can be used. However, as we will see below, keeping track of the size and position of the microparticles may be a useful means of keeping track of the growth of the different parts of the particle. We will, therefore, use the full MGM as a basis for our modelling, even though we know that the microparticle gradients are probably negligible. It should be pointed out here that this model has its limitations, and certainly cannot be applied to all types of catalyst/polymer systems (e.g. ). Nevertheless, it is reasonable to use the model developed here as a basis for exploring the influence of the relationship between the build-up of hydraulic forces and particle morphology since it can be modified if other particle models become available.One of the common assumptions when using the MGM and PFM is an instant and complete fragmentation of the catalyst particle. This assumption of complete fragmentation has been verified on MgCl2-supported Ziegler–Natta catalyst (). However, the initial rate of polymerization has been shown to have a strong influence on how fast a particle breaks up, which in turn, of course, influences how fast the reaction proceeds (). A more robust and physically realistic model of polymerization and particle growth requires a more thorough treatment of the breakup process. The first model to treat the topic of initial particle fragmentation in detail was developed by . In their work, they let the catalyst fragment shell by shell, with fragmentation occurring when the pressure in the polymer-catalyst particle, defined as a growth factor of the fragments, exceeded a pre-specified limit. They showed how this fragmentation could affect the reaction rate and the properties of the final product, but did not consider the possible changes in the macroscopic morphology of the growing particle. Their work also revealed that fragmentation is extremely rapid under most conditions, and it is, therefore, reasonable to use the assumption of instantaneous fragmentation as well.Another important characteristic of the MGM and PFM is the assumption of constant and uniform porosity during the growth process. There have been a few attempts at refining this. For instance, allowed the porosity to change as a function of radial position during particle growth. According to their model, if the outer parts of the particle grow faster than the core, the porosity increases in the outer parts since the core cannot follow that expansion. However, what is normally observed in homopolymer products is that the porosity slowly decreases with polymerization yield. This effect was implemented in an MGM-based model by by allowing the microparticles to become compressed and non-spherical in an empirical manner.Typical reaction rates of modern catalysts are much higher than those common when most of the existing models were developed . As the group of McKenna and Spitz have shown in a series of papers (), the traditional MGM and its implicit hypotheses do not function well for the higher activities common in industry today. The MGM overpredicts the significance of mass transfer resistance and cannot adequately model experimentally observed results. suggest a closer link between morphology and mass transfer as the key to understanding the process better. Also suggest that more adequate models of particle morphology are needed to explain their observations in experiments with highly active catalysts. For instance, they observe that the morphology changes with activity. As will be shown below, connecting mass transfer resistance and morphology in new ways leads to predictions of higher transport rates and better agreement with experimentally observed results.Despite the shortcomings cited above, the MGM is used as the basis for our model for reasons that will become apparent as we present its development below. Even though the microparticle concentration gradients normally are negligible, we will retain them in our model since they are a useful way of keeping track of the growth rates in different parts of the particle. In addition, in order to focus on the development of particle morphology, we will restrict our treatment to isothermal conditions. We, therefore, need only two pseudo-steady state material balances, one for the microparticle and one for the macroparticle, to calculate the concentration of monomer as function of position in the growing polymer particle. The reaction rate is calculated as in the MGM to determine the particle growth rate and evolution of the morphology through evaluating the induced tension in the particle. We call the new model the Tension model.Assuming pseudo-steady state and that the first order reaction takes place on the surface of a spherical catalyst fragment surrounded by polymer, showed that the reaction rate per mass of catalyst iswhere the effectiveness factor, η, is defined aswhere α=rc(kpρc/Ds)0.5 and φ=rs/rc and rs and rc are the microparticle radius and catalyst fragment radius, respectively. The reaction rate in terms of (mol/s) for one microparticle isThe rate of growth of the microparticles can then be derived asThis equation is solved together with the differential equation for the macroparticle growth as discussed below., the equation for mass transfer into a polymer particle should include a convective term:Here Ni is the molar flux of component i relative to stationary axes, c is the total concentration of molecules, D is the binary diffusion coefficient, xi is the mole fraction of the component and Ntot is the total molar flux, i.e. the convective transport of components. In the current paper, we limit our investigation to slurry polymerization since it is in this system one normally expects to encounter the largest monomer gradients at high activities. It is, therefore, reasonable to use an isothermal approximation, and to ignore the contribution of convection to mass transfer since the mole fraction of monomer is very small. The monomer flux in the particle can then be calculated fromThe macroparticle equation must be solved numerically and needs to be discretized. Instead of deriving the general equation and then discretizing it, we will directly derive the discretized equation. The method of discretization is the finite volume method as described by Normally, one can invoke the quasi-steady state assumption (QSSA) for monomer concentration in the particle (). The monomer concentration can then be found by solving the diffusion-reaction equations without considering the effect of particle growth. This assumption was verified by comparing the time scales for diffusion (R2/D) and particle growth. Using typical parameter values for the cases in this work (see ), yields a time scale for diffusion of less than a minute. The time scale for diffusion will depend on the media (gas or slurry) and particle size. Since the time scale of particle growth is minutes to hours the QSSA is valid in most cases (except polymerization with very large particles in slurry). For the initial part of the growth or breakup of catalyst, the time scale are fractions of a second for mass transfer and second for the growth or breakup. Based on these results, we think that the QSSA should give reasonable results.Applying the QSSA and doing the mass balance over a shell in the particle giveswhere Vc,i is the volume of catalyst in shell i and the indices are defined in . Vc,i is calculated using the catalyst particle size, porosity and density. for the flux of monomer in a dilute system givesThe concentrations are calculated in the centre of the shell and assumed to be representative throughout the same shell. When approximating the derivative at the shell interface, the distance between the centers of the shells, denoted dnode, is used as follows:The number of microparticles in shell i isSubstituting for the surface area of the shells results inThe system of equations can be reduced to a tri-diagonal matrix, which can be solved by a tri-diagonal matrix algorithm (TDMA) (The rate of growth of the macroparticle layers is the key to determining the evolution of the morphology. We can only do this if we can link the growth rates of the microparticles in one shell to those in another and understand how the growth rates influence the distribution of forces in the particle. In general one may write:As mentioned, one common assumption in the MGM is macroparticle growth with constant porosity. When the microparticles grow it is assumed that they are reorganized in such a way that the porosity does not change, even in the presence of gradients of monomer concentration in the macroparticles. In this case, the growth of the macroparticle layers is given byIn the examples that follow, we use predefined initial conditions to solve the ODEs using a variable order ordinary differential equation solver (ode113) in MThe equations above also form the basis for our new model. However, it is modified in such a way that we include the possibility that the particle can break when large enough tensions are generated in the growing particle.The growth of a portion of a polymer particle results in local volume change, and thus a localized stress in the material. We need an expression for this stress to examine what impact it can have on the morphology.The fragmentation of the catalyst and the growth of the polymer particle take place by alternating build-up and relaxation of elastic tensions in the polymer/catalyst particle. The formation of polymer causes expansion of the material and an increase in local tension. If the tension is large enough, the material will rupture.The following derivation is based on an example given by , who considered the stresses resulting from thermal expansion in a sphere subject to a temperature gradient. We have transferred this to the analogous situation that occurs when a polymer particle grows.We start by expressing the strains resulting from an expanded volume due to the polymerization. Consider a small control volume V0. If this control volume could expand without any external or internal stresses, the polymerization would cause it to expand to a new volume V. The strains that would be provoked by this expansion are:where εii is the relative elongation in the i direction. For small perturbations (i.e. ε≪1) this equalsSince we assume an isotropic material, the strains are equal in all directions and we haveConsider a sphere at rest. The momentum balance is simply:where the total stress tensor can be decomposed as σ=−I·p+τ. Assuming spherical symmetry and invariant pressure, the equations of momentum are reduced to only one:where τr is the radial component of the stress vector (= the diagonal of the τ tensor), and τt is equal to the tangential components (=τθ=τφ). The stress–strain relations are (Here λ and G are Lamé's elastic coefficients and E and ν are Young's modulus and Poisson's ratio, respectively. The first two terms in reflect the stress that one would observe if the polymer was deformed (e.g. stretched) with strains as εr and εt with no growth in the material. However, in practice some of this strain is a result of the growth. This must be subtracted in order to obtain the correct resulting strain that is the source to the stress. Thus the last term in We define a displacement function for the displacement of points in the sphere.where ρ is the position after deformation of the point originally located at r. The radial and tangential deformation can be expressed in terms of this function as, and further substituting into the momentum balance, where C1 and C2 are constants of integration determined by boundary conditions. Inserting this expression for U into the equations for the stress components, , yields the following relation between growth factor Φ, and the stress components:The equations above are the general equations for the stress components in a sphere subjected to an internal growth factor symmetrically distributed with respect to the centre of the sphere. Boundary conditions are needed to obtain more specific expressions. At the boundaries of the sphere, i.e. both on the outer surface and if relevant on the inner surface (hollow particle), the momentum balance over the boundary yields:In case of a solid sphere, the displacement in the center is zero, givingIt turns out that the last boundary condition () will be the equivalent to the case of allowing the inner radius of a hollow sphere to approach zero and let τr(r0)=0. Thus it is possible to formulate general solutions that cover both cases of hollow and solid spheres. Applying the boundary conditions to evaluate the integration constants, gives the following expressions for a (hollow) sphere with inner radius r0 and outer radius R:We tested these equations on some simple model cases. A constant Φ, equivalent to uniform growth in the particle, gives τr=τt=0. For compact particles with the growth factor Φ being a steady rising function with radius, as expected when monomer concentration gradients in the particle are present, the largest radial tension will be found in the center of the particle. This can lead to the formation of hollow particles as will be shown later. It is also found that in this case the tangential stress is equal to the radial stress in the center and shifts sign towards the surface of the particle. Thus, at the surface there is a compressive stress in the tangential direction. For the case of a Φ that decreases toward the outside of the particle (core expands more than shell), a stretching tangential stress is found near the surface of the particle. Thus the model shows the expected results.We need to relate the growth factor to rate of polymerization in order to predict the tension in a growing polymer particle. As stated in the preceding section, the growth factor is determined by the relative volume change when there are no stresses present. The production of a volume Vpol of polymer (not including pores) in a volume Vc of catalyst (not including pores), yields the growth factorwhere εp and εc are the porosities in the polymer and in the catalyst particles, respectively. The volume of polymer is found by integrating over the time of polymerizationwhere Mc is the total mass of catalyst. Using in the stress relations above would return incorrect results since they are developed for rigid bodies, and because the stress equations were derived for small perturbations only. During polymerization, the volume changes are in the order of thousand. Secondly, the polymer material is viscoelastic, thus the stresses are eventually relaxed. To account for the viscoelastic effects, one needs a viscoelastic model. We use a more simple approach here by proposing some approximations, but will retain the important features of the model.In the model, growth takes place in a pseudo-steady state manner, and the build-up and relaxation of tension are approximately equal. Since the stresses are relaxed, the denominator V0 in the growth factor expression in can be approximated using the value of the present volume of polymer in a shell, since the volume V0 is the “stress free” volume and only small perturbations are assumed to cause the stresses. Further we assume that the volume change, ΔV, can be set proportional to the volume of polymer produced per unit time, i.e. proportional to the reaction rate. The faster the reaction, the larger the perturbations will be relative to the stress-free configuration. The equilibrium between build-up and relaxation of stresses is thus assumed to be proportional to the reaction rate. With these assumptions, the modified growth factor can be written asSubstituting this in the stress relations, where E′=E/(3(1−ν)). For dVpol and V we use the concentric layers as basis. The volume of polymer formed in layer i is thusImplicit in the assumptions above is that any instantaneous change in porosity is small. A rigorously correct model would of course include this variation as well. However, as we discussed above, this change is generally thought to be very slow. It is possible, particularly in the early stages of polymerization (first few seconds), that this can have an effect. However, in this initial development we will not try to model simultaneous rupture of the catalyst matrix and of the polymer matrix, and content ourselves with trying to describe the polymerization following catalyst breakup, where the assumption that porosity does not evolve has been shown to be a good assumption.We define a model parameter τlim as the limiting tension that a matrix of a growing particle can undergo without rupturing. If the tension inside the particle exceeds this value at a given point, the particle will break at this point, and the layers on either side of it will separate. In the present model we only allow for separation in radial direction and we do not consider effects of compressive tensions. The implementation of this model is shown below.Assume that the radial or tangential tension exceeds τlim at i=b (i.e. the particle ruptures at layer b). In this case, the outer layer n expands freely—in other words, the relative growth of layer n and that of the microparticles are equal, and the layers b…n−1 follow the outer layer by growing with a constant porosity:The remaining core (layers i<b), is assumed to grow with constant porosity.For a homogeneous particle (evenly distributed catalyst fragments and amount of polymer), the lowest monomer concentration is at the center. This means that the reaction rate and the volume expansion are lowest here. In this case φ increases monotonically with the radius, and the largest stretching tension will be found in the center of the particle. Thus at high activities, the most common result of the Tension model would be a breakpoint right at the center of the particle and the formation of a hollow particle. On the other hand, when monomer and catalyst concentrations are constant throughout the particle, the growth factor will be constant and no tension will be induced. In this case, the basic model (i.e. the MGM) predicts that the particle will grow uniformly (expansion with constant porosity). Thus, in the absence of mass transfer resistance we expect no particle rupture.The particle rupture of a growing polymer particle will not only have an impact on the morphology, but will also have great influence on the mass transfer resistance. Rupture will increase the outer radius of the particle. The outer surface of the particle increases and the distance between surface and the inner polymer shortens. The increased surface area and the shortened diffusion length scale will lead to a reduction in mass transfer limitations.As we will show below, large mass transfer resistance leads to large tensions in the growing polymer particle, which in turn leads to rupture and decreased mass transfer resistance. Thus a growing polymer particle has a sort of self-regulating mass transfer resistance property. This should not be surprising since one almost never observes really dramatic effects of mass transfer resistance in polymerizations with new effective catalysts.In this section we will compare model predictions from the MGM to those obtained from the Tension model derived above to demonstrate the potential influence of reaction kinetics on particle morphology in the slurry polymerization of ethene. The simulations are compared to two experimental case studies. Model kinetics are adjusted (see below) in order to approximate the rates of polymerization in the experiments.Case A is a laboratory-scale study reported by where the original goal was to measure the polydispersity as function of activity using a TiCl4 Ziegler–Natta catalyst supported on MgCl2. Peak polymerization rates were between 5 and . Case B is a series of polymerizations taken from a study by Borealis AS on the impact of activity and kinetics on the morphology of particles produced on metallocene catalysts. Two different catalyst complexes were supported on two different support sizes (i.e. four different catalysts). Both cases were polymerizations of ethene in a solvent. The kinetics of both systems were described empirically with the following expression for the activity:where the deactivation function is given byand H(t) is the step function. The basic parameter values are given in For case A, the polydispersity was calculated assuming that the number average molecular weight is directly proportional to the monomer concentration, and that the intrinsic polydispersity of the active sites is equal to q0=8.0 (see Both cases described above produce distinct variations in morphology with reaction conditions. Generally, both cases show the appearance of hollow particles in cases where reaction conditions are expected to provoke the highest mass transfer resistances (i.e. large catalyst particles, high intrinsic activity).We examined polymer particles from Mattioli's series (2000) (case A) by SEM. At low activity , it was found that the particles showed a randomly distributed porosity, with fairly uniform internal morphology and an average void fraction of about 25%, whereas the samples from the high activity runs contained hollow particles, pieces of shells and smaller fragments. Examples of SEM pictures from these series are shown in In case B, the kinetics of the two catalyst complexes were quite different. The more active one had a sharp peak in activity with a maximum at about early in the polymerization (time), leveling off at about . The other complex showed a monotonic increase in activity to a stable level of about . SEM pictures of whole particles and cross sections of particles embedded in an epoxy resin show that hollow particles are formed only when using the largest of the two catalyst supports together with the more active of the two catalyst complexes. The other three combinations of support and catalysts showed a more compact morphology. This is also supported by the bulk density of the samples. Pictures of particles from case B are given in As shown above, more hollow particles are formed when the largest mass transfer resistance is expected for both systems. More compact or uniform particles are formed when less severe monomer gradients are expected. To be able to predict this, physical limits for the breakup of the particles need to be identified.The tension limit will, in general, depend on the porosity and friability of the catalyst support, the material properties of the polymer (crystallinity, molecular weight, softening point etc.) as well as on the conditions in the reactor (temperature, pressure, phases, etc.). At the current time, we are not trying to develop an exact relationship, but rather are only trying to explain and predict trends in order to identify the most crucial model parameters and assumptions. For this reason, we use a constant tension limit in the simulations, and define distinct values for each of the two cases A and B.Since the elasticity parameter E′, is an unknown parameter we define a reduced tension and tension limit according toOne way to find the tension limit is to simulate cases with conditions which are similar to the ones where ruptured particles start to show up experimentally, and to use these values to obtain an indication of the limiting tension for breakup.For case A, the average activity where hollow particles started to show up was at about . Simulation with this activity and the parameters as in For case B, the critical tension limit was assumed to be reached with only the largest catalyst particles with the high activity complex. Simulations gave There are several explanations for these observed differences. Since the two catalyst complexes are different, there can be large differences in the polymer properties (crystallinity, molecular weight, etc.) and thus the tension limit. Also the catalyst supports are very different. In case A, magnesium chloride was used as the support, and in case B it was silica. This can have an impact both on the breakup of the catalyst particle, and on the final morphology (e.g. size of polymer domains). We also assumed that the spatial variation in expansion rate to be only a result of monomer concentration gradients. However, a possible non-uniform distribution of catalyst can give the same effects. And finally, we do not account for initial inhomogeneities in the catalyst support (large pores, cracks) that can also have an impact on the fragility of the particle.We simulated the growth of a single particle for different activity levels using the parameter values for case A. To compare cases with and without rupture of particles, we simulated a case with a low activity (peak intrinsic activity about ) and one with high activity (peak intrinsic activity about ). The intrinsic and simulated reaction rates as function of time are plotted in and snapshots of monomer concentration gradients are plotted in At low activity, the tension in the particle is smaller than critical breakup tension, so both the MGM and Tension models predict growth of particles with constant porosity. The mass transfer resistance is significant, but the difference between simulated overall activity and intrinsic activity is not more than 25%. The largest monomer concentration gradient occurs early in the polymerization at about At high intrinsic activities, the resulting concentration gradients cause the tension in the particle to rise above the critical tension limit, and it can be seen that the two models predict very different results. First of all, “observed” activities with the MGM are much lower than the intrinsic levels seen in . This means that the monomer concentration drops to zero a short distance from the surface of the particle, and the inner parts do not grow at all. The Tension model, on the other hand, predicted the formation of hollow particles for an “observed” activity of and the tension exceeded the critical tension defined above. This was a result of increased tension at the center of the particle and a subsequent rupture of the matrix. It is important to differentiate this from the hollow particles some authors predict using the MGM. In the case of the MGM, this is entirely due to lack of monomer at active sites toward the center of the particles. The active sites in the monomer poor regions do not participate in the reaction and the activity in decreases. In the Tension model, the hollow center contains no active sites, and all of catalyst is used for the reaction. However, the rupture allows the particle to expand which means that the surface area available for mass transfer with the bulk phase is higher than for the MGM and we can supply all of the active sites with more monomer. If we look at , we can see that the monomer concentration at the rupture point in the Tension model (r/R=0.75) is much higher than the concentration at the center of the particle for the MGM at high activities. This means that the average monomer concentrations for the Tension model are much higher than for the MGM.The polydispersity index, q, was calculated as function of observed activity. It was assumed that the chain termination rate was independent of monomer concentration (i.e. that hydrogen dominated the chain termination reaction) according to the method described by . The polydispersity was measured experimentally by The Tension model shows only slight variations in polydispersity with activity, whereas the MGM predicts that the polydispersity increases significantly due to the presence of large monomer gradients. The experiments show no significant variation in polydispersity with activity, which suggests that the MGM significantly over-predicts mass transfer resistance. For low activities both models provide satisfactory agreement with experimental data.In the Tension model, the activity level corresponding to the point of rupture (i.e. the activity at which the tensions exceed the predefined tension limit) is reflected by a sudden change in the predicted polydispersity in . Obviously, we do not expect to detect such a change experimentally where a range of particle sizes, variations in geometry and eventual small chemical differences between the particles will mask the sudden transition the curve.For case B, the Tension model was used to stimulate particle growth for four combinations of catalyst sizes and complexes. The experimental and simulated activities as function of time are plotted in and the corresponding monomer concentrations as function of particle radius are shown in The intrinsic activity was varied until the predicted activity coincided with the experimental values. The concentration plots are snapshots after of polymerization, which corresponds to the time where the largest concentration gradients are present. It can be seen that the monomer concentration is, on average, higher for the case where the particle has broken (large particle with high activity) as was the found for case A.The concentration gradients in case B are much lower than what was found in case A. The experiments show that the particles break and thus the fitted tension limit in case B is much lower (a factor of 10), and the model predicts particle rupture at much lower concentration gradients. Note that a non-uniform active site distribution with a higher site concentration near the surface could provoke the same effects as a monomer concentration gradient. Given the large difference in the predicted value tension limit for the two cases, we cannot exclude the possibility that the active sites are not uniformly distributed for catalyst system B.The Tension model seems to provide an explanation based on the evolution of the particle morphology of the different catalysts systems used in the work that agree with experimental observations of morphology and polymerization rate. Small particles, and weak concentration gradients (generally from low activity) favour the production of “solid” particles. Large particles and strong concentration gradients can provoke particle rupture. This is summarized in It is important to point out the limitations of the model presented above. The simulations show that the highest tension is obtained very early in the polymerization for both case A and B, with the maximum value being reached after a few tens of seconds from the beginning of the reaction. At this point, the productivity is low (on the order of a few grams per gram of catalyst). It is well known that the catalyst goes through a breakup process at this stage in the polymerization, and we have, therefore, over-simplified the complex physical processes involved by assuming instantaneous fragmentation. Nevertheless, the results of the simulation clearly show the relationship between activity and the evolution of particle morphology—which is the principal objective of the current study. In fact, a two-staged model that applied a similar treatment to both the fragmentation process, and the particle growth process with tlim being set differently for each stage might be even more realistic.The different tension limits we found for the two cases above also indicates that the breakup of the two different catalysts might be the key process in the determination of the final morphology. Given the experience that e.g. has on the importance of the initial period of polymerization on the final morphology. It is clear that the emphasis in future work should be on the evolution of particle morphology during the early stages of fragmentation, eventual rupture and particle growth.Commercial production of polyolefins faces demands like high activity, easy post-process treatment and good morphology of the product. A good particle model is vital in the struggle to optimize the process with respect to these demands. It would be useful to be able to manipulate particle morphology in order to better control the properties of the polymers being produced (thus their end-use value). For instance, have all stressed the role of particle morphology in producing multilayered olefin products (e.g. high impact PP). have shown that mass transfer limitations can be linked to particle morphology in the production of such products. Also, the formulation of hollow and ruptured particles is an undesirable effect. These particles have less resistance to external mechanical forces and can easily disintegrate during reaction and post-reaction handling operations. Finally, particles that rupture internally will have higher porosity and much lower bulk densities. The Tension model can be used to determine what tension will be present in a growing polymer particle. To avoid the formation of many internal ruptures, one needs to keep the internal tension below a critical value. However, to obtain high activity with large particles, one should aim at operating close to this critical value.Also the effects of alternative designs of catalytic systems could be investigated with the Tension model. For instance, one could imagine compensating for monomer concentration gradient by changing the concentration of active sites at different spots in the particle. Also prepolymerization would clearly help to decrease the tension in the particle, and the model can be used to find to what extent and under which conditions the prepolymerization should be done.A new single-particle model, the Tension model, was developed and used to simulate the growth of polymer particles in two case studies. The model couples activity, mass transfer resistance and the development of morphology. It was demonstrated that a spatial variation in the growth rate of polymer caused by monomer concentration gradients might lead to a spatial variation in the expansion rate, and thus the build-up of elastic tensions inside the growing particles. If these tensions exceed a pre-determined limit, the particle undergoes internal rupture.In reactions where high intrinsic activities and/or the use of large particles lead to the formation of concentration gradients due to mass transfer resistance, the internal stresses will be higher than for low activity catalysts. Particles are more likely to rupture in these cases, and if the particles are internally homogeneous, they will rupture at the center. This can lead to the formation of hollow particles that expand more rapidly than particles without rupture. This decreases the mass transfer resistance, and therefore, leads to the formation of much lower concentration gradients than predicted with the MGM. Thus the Tension model predicts a self-regulating property of the system.Finally, the largest tensions arise after a few seconds of polymerization and the catalyst goes through a breakup process at this stage in the polymerization. The different tension limits found for the two cases studied also indicates that the breakup of the catalyst is important when determining the final morphology. Future work in this field should be focused on the catalyst breakup process.Tribology Research: From Model Experiment to Industrial Problem G. Dalmaz et al. (Editors) .9 2001 Elsevier Science B.V. All rights reserved. The prediction of sliding wear: theory, experiment and finite-element analysis Y. Blake Department of Mechanical Engineering, Institute of Technology Tallaght, Dublin 24, Ireland. When one metal slides across another, plastic strains are introduced into the surface, these strains being related to the detachment of wear particles. A quantitative law of wear rates has been established through the extent and rate of change of strain due to the sliding. A series of experiments was performed on non-ferrous metals. Particular changes found in the microstructure revealed that the wear damage is a continuous process. Finite element analysis was applied to simulate the wear process. Results are presented for the comparison of the theoretical approximation and experimental measurements. An attempt is made to relate the bulk mechanical properties of the material to the response to the wear. 1. INTRODUCTION When one metal slides across another softer one, plastic strains are introduced into the surface of the soft metal and this strain is related to the detachment of wear particles. The establishment of a quantitative law of wear rates has been made through the extent and rate of strain due to sliding. Several authors have developed models, which allow the calculation of wear coefficients from the strain cycles induced by the asperity interaction [1- 3]. One of the main difficulties encountered in the wear prediction has been to model realistically with analytical approximations the actual strains around an asperity. In the case of a large strain field, the rigid plastic model [4] has proved capable of predicting wear. The experiments of various workers have shown that the rigid-plastic model can predict friction well for sliding asperity experiments [5-7]. However, in the situation when plastic and elastic strains of similar size coexist, an analytical model is hard to find. The few attempts that have been made to produce such a model [8-10] have shown the importance of this part of the field in determining the direction of crack propagation, the shape and size of wear particles and the prediction of wear rates. With the more powerful finite element analysis (FEA) packages currently available it has now become feasible to use such methods to study asperity contact problems. In our previous work [11 ] wear was assumed to be a discontinues process. There were also difficulties to obtain the wear data from copper. Here a more detailed investigation of wear processes is presented by using the same single asperity interaction, but in which the slip-line field model and finite element analyses were applied. Experimental tests were conducted to verify the results of the calculation. The primary equations of the rigid plastic models that were used will now be listed. Calculation of the coefficient of friction It: A sin ct + cos(arccos f -or)/t = (1) A cos ~ + sin(arccos f -ct) where the independent variables are the angle of the hard asperity ot and the Tresca factor f= 'r,/k s with x the shear strength of the film and k s the shear yield strength of the deforming material. The number of asperity passes required to produce a wear particle N is determined by the relation: (old N f= Are (2) where C is the monotonic effective shear strain for fracture in one cycle and D is the exponent in the Manson-Coffin fatigue law, where both must be found experimentally. Aye is a global plastic strain induced in the surface by the passage of one asperity. The wear coefficient K is expressed as a function of the properties of the contact: K= 9~r3ru (3) cD --1-D AT" ~ where/t is the coefficient of friction, r the ratio of plastic work in the surface to the total work of sliding. The plastic strain Ate can be found from energy considerations. A knowledge of the wear coefficient K allows the Archard equation to be used for wear prediction: KNL V (4) - 3H where V is the volume worn from a material of Vickers hardness H after sliding a distance L under a normal load N. 2. THE EXPERIMENT In the experiments a hard asperity is indented and slides on the outer surface of a rotating disc/bar of the relatively soft specimen material. This is ~ Asperity L Soft Material shown in Figure 1. Figure 1. The experimental set-up. A series of tests were performed with a range of asperity angles on aluminium, brass and copper. The reason for the selection of these metals is that they have similar harnesses but different ductilities. Wedges of different angles (a) are used to represent changes in surface roughness. The wear of the disc or bar is measured by weighing before and after a test. During the test the normal indentation force and the frictional force are measured thus allowing the calculation of the friction coefficient/t. When the test is run for a sufficient number of revolutions the weight loss can be measured as overall wear and the experimental wear rate can be determined. In all cases the strain AYe per cycle and the number of cycles to failure Ny were found. Finally the predicted wear rate using the rigid-plastic model can be compared with the experimental wear rate. The finite element software used for the analysis was MARC 320. The asperity was assumed rigid. Contact elements were used to define the interracial conditions between rigid asperity and the soft metal. 3. RESULTS AND DISCUSSIONS 3.1 The Friction Coefficient The theoretical friction coefficients/t calculated from equation (1) are plotted against the asperity angles a shown in Figure 2 as dashed lines. 0.6 t't FEA f=0.20 I AI A Cu f .9 f=0.10 Jt I/ /~/ f=O 0.4 .* I~.~ / A 0.2 .9 . 0 -- 0 10 20 30 Figure 2. Friction coefficients: Slip line field prediction (---), FEA calculations (), Experimental measurements of aluminium (*) and copper (A). with the rigid plastic model. Thirdly, the diagram shows that strains associated with higher wedge angle fall faster towards the bottom of the deformation zone; as a result, the thickness of the deformed layer is thinner than the deformed layer produced with the smaller angle. All of these coincide with the observation from the wear test specimens. As was mentioned before, the slip-line field theory assumed that no wear had taken place until the deformed layer h had suffered sufficient strain throughout its volume for failure to occur and to be removed. In this way the theory describes wear as a removed a layer with thickness h. Within this deformed layer the strain is homogenous. Both experimental observation and FE analysis suggest that wear is a continuous procedure rather than a discontinuous one. Soft material is severely deformed at the wear surface and becomes steadily less deformed as it approaches the bottom of deformed layer. The layer that was actually removed from the worn surface due to each wave pass is much thinner than the thickness h of the deformed layer. Once the wear starts it is progressed continuously. leo ~11 15 ~ O---~ 15~ II m --II 20 ~ 0---0 200R 0.9 o 'R .0..~~O~ " ~ o.o, ,,'m ~ "~" 0.9 "Jlll~ ~' "" "'" e=-"=~_ o ' * 0.4* ' 0.8 * 0.2 * * 0.6 1.0 Distance from surface Figure 6. Comparisons of total equivalent strains of sharp and rounded wedges of 15 ~ and 20 ~ . For the situations in which the tip of the wedge became rounded the values of the total strain predicted by the FEA are greater near the surface but drop faster than the sharp wedge as the bottom of the deformed layer is approached, as shown in Figure 6. The calculations provide a better explanation of the severe deformation near the surface and why the 181 strain appeared to be approximately constant in this thin layer. If the overall strains induced into the surface can be estimated by the area under the profile of the total stain, from the same diagram it can be seen that the overall strain induced by the rounded wedge is less than the sharp wedge. This could be the reason why a blunt wedge produces less wear. 3.3 Wear Coefficient K and Properties Having determined the values of C and D from the analysis of the wear results and A~ through effective energy method, the theoretical wear coefficient Kth is calculated using equation (2). The values of the wear coefficient K~xp obtained from experimental results were calculated using the following equation: Kexp = O'yV LN (7) where Cry is the yield stress of the soft metal, V is the wear volume, L is the sliding distance and N is the normal load. 1 ' ' I ' ' ' ' r t I I I I I I i I i I I I L.M" ~ I 0, -~ ,--t--r--i~F~*-........ -1 ! = , / O.Ol I I I , o.ool . . . . . . . . . . . -.-I- I 0.0001 . . . . --I ,~ i,(I ~,, I ,I I I I I I I I I I I I I I I I I I I I I I 0.00001 I I I I I l I 2 5 10 20 Wedge angle (x Figure 7. The predicted ( .... ) and experimental wear coefficients (,O,A) of three soft metals. In Figure 7 the experimental wear coefficient Kex p obtained from wear tests on three metals are plotted in comparison with the theoretical values Kth predicted by rigid plastic model. It can be seen that the agreement between experiment and prediction are reasonably good. From the results presented it may be concluded that for the three materials tested the rigid plastic model is able to give an upper limit of the wear coefficient when the mechanical properties, surface finish and lubrication conditions are given. The actual wear coefficient under the given conditions will not exceed this estimation when the total number of wedge passes R is close to, or greater than the number to failure N/ for copper and 3N: for aluminium and brass. The mechanical properties of these metals together with the material constant C, the number of cycles to failure N/and the wear coefficient K for asperity angle of 15 ~ are given in the following table: Brass Aluminium Copper Hv 170 125 116 277 MPa 125 MPa 116 MPa 0.21 0.69 2.04 C 6.26 9.96 26.0 N: 4.8 105. 349.5 K,h 3.949x10 -1 1.551x10 -1 6.382x10 -3 * el=-ln[(100-R.A.)/100], R.2L = Reduction of Area The above results show that the material with higher ductility gives a higher number of cycles to failure N I and a lower wear coefficient K. The number of cycles to failure N I is therefore related to the mechanical properties of the material, especially the ductility. It also suggests that the effective ductility for wear particle formation may be simply related to strain to failure in conventional mechanical tests. 4. CONCLUSIONS The study presented here is in its preliminary stage. The results assume that the rigid-plastic approximation together with experiments and FEA make it possible to predict wear. There is very good agreement between the prediction of the friction coefficients by rigid plastic model and FEA calculations. The experimental measurements show an increasing departure from those predicted when the attacking angle a is greater than 150 . The effect of the entrapment of wear particles is used to interpret this departure. The influence of the third body on the wear rate should therefore to be further studied. The FEA calculations provided detailed information on strains induced through wear which the rigid plastic model could not predict. It showed that the total equivalent strains vary with the depth in an exponential manner. The examination of the worn surface revealed the same features. Both suggest that wear is a continuous procedure. FEA results also gave the changes in strain when a sharp wedge became rounded, a situation which occurs in reality, but is difficult to explain through analytical approximation. There is an attempt to relate the mechanical properties of the soft metals to their response to wear but further work is needed to establish a quantitative correlation. The single asperity model has the potential to be extended to a multi-asperity surface: Moalic et al. [13] in considering friction and Lacey [6] in considering wear have done this successfully. Further investigations should be carried out in this direction. A combination of theoretical and experimental approaches is helpful as they may lead to quantitative understanding of the mechanism of wear and to the prediction of parameters describing the wear phenomenon. REFERENCES 1. J.M. Challen, P.L.B. Oxley and B.S. Hockenhull, Wear, 111 (1986) 275. 2. P. Lacey and A. A. Torrance, Wear, 145 (1991) 367. 3. A.A. Torrance and A. Parkinson, 19 th Leeds- Lyon Symposium, Ed. D. Dowson and M. Godet, Elsevier (1993). 4. J.M. Challen and P.L.B. Oxley, Wear, 53 (1979) 229. 5. J.M. Challen, E.M. Kopalinsky, P.L.B. Oxley, Tribology- friction, lubrication and wear fifty years on, Vol. II, I.Mech E., London Paper C156/87, 1987. 6. P. Lacey and A.A. Torrance, Wear, 145 (1991) 367. 7. W.H. Roberts, Int. Conf. on Tribology, Tokyo, 1985, Japan Society of Lubrication Engineers 8. A.V. Oliver, H.A. Spikes, A.F. Bower and K.L. Johnson, Wear, 107 (1986) 151. 9. A.A. Torrance, Proc. I. Mech E., 208 F (1994) 113. 10. A.A. Torrance and F. Zhou, 20 ~ Leeds Lyon Symposium, Ed. D. Dowson and M. Godet, Elsevier (1994). 11. Y. Yang, A.A. Torrance and P.L.B. Oxley, J. Phys. D: Appl. Phys. 29 (1996) 600-608. 12. M. Busquet PhD thesis, Trinity Collelege Dublin, 2000. 13. H. Molic, J.A. Fitzpatrick and A.A. Torrance, Proc. Instn. Mech Engrs. 201 (1987) 321-329 onditions are given. The actual wear coefficient under the given conditions will not exceed this estimation when the total number of wedge passes R is close to, or greater than the number to failure N/ for copper and 3N: for aluminium and brass. The mechanical properties of these metals together with the material constant C, the number of cycles to failure N/and the wear coefficient K for asperity angle of 15 ~ are given in the following table: Brass Aluminium Copper Hv 170 125 116 277 MPa 125 MPa 116 MPa 0.21 0.69 2.04 C 6.26 9.96 26.0 N: 4.8 105. 349.5 K,h 3.949x10 -1 1.551x10 -1 6.382x10 -3 * el=-ln[(100-R.A.)/100], R.2L = Reduction of Area The above results show that the material with higher ductility gives a higher number of cycles to failure N I and a lower wear coefficient K. The number of cycles to failure N I is therefore related to the mechanical properties of the material, especially the ductility. It also suggests that the effective ductility for wear particle formation may be simply related to strain to failure in conventional mechanical tests. 4. CONCLUSIONS The study presented here is in its preliminary stage. The results assume that the rigid-plastic approximation together with experiments and FEA make it possible to predict wear. There is very good agreement between the prediction of the friction coefficients by rigid plastic model and FEA calculations. The experimental measurements show an increasing departure from those predicted when the attacking angle a is greater than 150 . The effect of the entrapment of wear particles is used to interpret this departure. The influence of the third body on the wear rate should therefore to be further studied. The FEA calculations provided detailed information on strains induced through wear which the rigid plastic model could not predict. It showed that the total equivalent strains vary with the depth in an exponential manner. The examination of the worn surface revealed the same features. Both suggest that wear is a continuous procedure. FEA results also gave the changes in strain when a sharp wedge became rounded, a situation which occurs in reality, but is difficult to explain through analytical approximation. There is an attempt to relate the mechanical properties of the soft metals to their response to wear but further work is needed to establish a quantitative correlation. The single asperity model has the potential to be extended to a multi-asperity surface: Moalic et al. [13] in considering friction and Lacey [6] in considering wear have done thisThe prediction of sliding wear: theory, experiment and finite-element analysisWhen one metal slides across another, plastic strains are introduced into the surface, these strains being related to the detachment of wear particles. A quantitative law of wear rates has been established through the extent and rate of change of strain due to the sliding. A series of experiments was performed on non-ferrous metals. Particular changes found in the microstructure revealed that the wear damage is a continous process. Finite element analysis was applied to simulate the wear process. Results are presented for the comparison of the theoretical approximation and experimental measurements. An attempt is made to relate the bulk mechanical properties of the material to the response to the wear.A conservation integral for arbitrarily curved interface cracks under thermal loadingA conservation integral, which consists of the path and area integral, for arbitrarily curved interfacial cracks under thermal loading is proposed and shown to have the physical meaning of energy release rate for the curved interface cracks. Three cases of interface cracks under thermal loading are investigated to show the usefulness and convenience of this conservation integral.Curved cracks are frequently observed at the curved interface between dissimilar materials, and/or when a material experiences mixed mode loading In order to make this paper self-contained, we reiterate briefly the previous results of the G* integral, which was proposed by Beom and Earmme where e3jk is the permutation symbol, W, the strain energy density of , ni, the unit outward normal vector, ui, the displacement vector, ti, the traction vector and ϕ, a continuous function up to the second derivative in A. Using Γ0, a closed contour as shown in along with the divergence theorem and the equation of equilibrium, it can be easily shown that G*=0 assuming no singularities in A. Suppose a crack along the curved interface given by ϕ(x1,x2)=0 in and the crack surface is traction free. It can be shown that the G* integral defined by is path independent for any Γ enclosing the crack tip. Area A is enclosed by Γ, Γ+ and Γ−, where Γ+ and Γ− are the paths at the upper and lower crack surfaces, respectively. It has been proven by Beom and Earmme where Γε is a circular path enclosing the crack tip with a vanishingly small radius ε. We normalize ϕ so that the magnitude of at the crack tip is unity and the direction of at the crack tip is along the x2-axis as shown in , is the normalized equation of the interface. Replacing ϕ(x1,x2) by , it is proven that G* integral corresponds to the energy release rate. Thus, the energy release rate of a curved interface crack is written asWe now consider a curved interface crack that is subjected to a two-dimensional small deformation field and is traction free under thermal loading as shown in . Under thermal loading, we can define strain energy density aswhere εij is the total strain including thermal strain of αθδij, α, the thermal expansion coefficient, θ, the temperature difference between the ambient and the reference temperature, and μ and λ are Lame’s constants. We define the G* integral under thermal loading asWe can easily show that G*=0 using the same method as described in the case of mechanical loading with no singularities in Γ0. under thermal loading and define the G* integral bywhere area A of integration is enclosed by Γ. This G* integral can be shown aswhere Γε is a circular path enclosing the crack tip with a vanishingly small radius ε. In obtaining , G*=0 with the fact that e3jkϕ,knj=0 along Γ+ and Γ− has been used. clearly shows that the G* integral, redefined under thermal loading is path independent of any Γ and A. Comparing , we can see that the G* integral under thermal loading has one more term, ∫Ae3jkW,θθ,jϕ,kdA, in addition to the original G* integral.Let us consider that a curved interface crack in is assumed to grow along the interface described by ϕ(x1,x2)=0. Since the equation of the interface ϕ(x1,x2) is not unique, we normalize ϕ(x1,x2) to make the magnitude of is the normalized equation of ϕ(x1,x2). Replacing ϕ(x1,x2) by is the unit tangent vector to the interface at the crack tip. The line integral in represents the energy release rate when the interface crack grows along the interface. Thus, the G* integral defined under thermal loading has the physical meaning of energy release rate. Consequently, the energy release rate, G for the curved interface crack under thermal loading can also be written asIt is clear that the value of the integral G in is independent of the size and shape of the integration domain.To illustrate the usefulness of the domain integral expression (10) for the energy release rate under thermal loading, we consider three examples assuming thermal insulation at crack surfaces: (1) a circular arc crack along a circle in an elastic solid, (2) a crack along an elliptic inclusion in an elastic solid, and (3) a cosine shaped interface crack along the bond line between film and substrate. The first two examples can be utilized in studying the fracture behavior of fiber (circular or elliptic) and matrix, and the third one is for the film debonding, or for the fracture behavior of semi-conductor packaging. Elastic analyses using regular eight-noded isoparametric elements are carried out with the is carried out in a separate post-processing program.Let us consider a semi-circular crack along a unit circle in an elastic solid under thermal loading of uniform heat flux at infinity as shown in The stress intensity factors were derived by Chao and Shen where μ is the shear modulus, κ=3−4ν, β=(1+ν)α, ν, the Poisson’s ratio, γ, the angle of the heat flux and x-coordinate, q0, the heat flux at infinity, and k, the thermal conductivity. Adopting the similar method of Li et al. α are evaluated at the nine Gauss points, ωα is the weight in the Gaussian integration and and (b). Four integration domains are chosen to calculate the energy release rate of numerically. The numerical values used in this analysis are: , E (Young’s modulus) = 200 GPa, ν=0.3, k=10 W/m °C, α=1.0×10−6/°C. shows the comparison of the energy release rates to the exact one through the heat flux angle from 0° to 360° and shows good agreement with them. The results in the case of heat flux angles 0° and 40° are compared to the exact energy release rates at crack tips A and B, which are calculated by and the comparison results are given in The most interior domain is 1 and the outermost domain is 4. This table shows that the G* integral is in good agreement with the exact values and has excellent path independence of the integration domain. In order to evaluate the sensitivity of the G* integral on the mesh density, we compare Gexact of with the results of coarse mesh of 64 elements and fine mesh of 400 elements around the crack tip, respectively. clearly shows that mesh density has little impact on the energy release rate, and energy release rates for each domain using fine and coarse mesh are constant within 1%. From this example, we can clearly understand that the G* integral has the property of good path independence and can be used to calculate the energy release rate of curved cracks.In order to verify the effectiveness of the G* integral for the curved interface crack under thermal loading, here we consider cracks along an elliptic inclusion in an elastic solid. An interfacial crack lies along the bonding line of an elliptic inclusion and matrix as shown in The material of the matrix is epoxy and the inclusion is carbon fiber. Numerical values used in this analysis are: shear modulus, μ1=124.0 GPa and μ2=1.24; Poisson’s ratio, ν1=0.2 and ν2=0.4; thermal expansion coefficient, α1=2.7×10−6/°C and α2=57.6×10−6/°C; thermal conductivity, k1=15.6 W/m °C and k2=0.45 W/m °C. Here, the subscripts 1 and 2 denote materials 1 and 2, respectively. The crack angle of 90°, a/b of 2 and the heat flux angle of 90° are chosen and the calculation results are given in shows the model for the cosine shaped interface crack that lies along the bond line between a film of copper and a substrate of epoxy. Here, the amplitude of the cosine shaped bond line of 0.5 and 1 are used to see the impact of the amplitude of the cosine curve on the energy release rate. Numerical values used in this analysis are Young’s modulus, E1=119.0 GPa and E2=166 GPa, Poisson’s ratio, ν1=0.34 and ν2=0.25; thermal expansion coefficient, α1=17.0×10−6/°C and α2=2.3×10−6/°C; thermal conductivity, k1=196.0 W/m °C and k2=153.0 W/m °C. Here, the subscripts 1 and 2 denote a film of copper and a substrate of epoxy, respectively. Temperature at the bottom is fixed at 100°C and the temperature at the top is fixed at 0°C. Analysis results are given in In these two figures, the energy release rates are normalized to the J-integral values that are obtained from the same length of a crack when a cosine shaped crack is projected to the x-axis and we can see that the energy release rate of cosine curve with the amplitude of 1.0 is greater than that of 0.5.A conservation integral for arbitrarily curved interface cracks under thermal loading, which consists of path and area integrals, is proposed by modifying the G* integral as suggested by Beom and Earmme Multidimensional force spectra of CNC machine tools and their applications, part two: reliability design of elementsThis paper deals with the reliability design of elements of computerized numerical control (CNC) machine tools and the practical applications of multidimensional force spectra of CNC machine tools described in part one of this paper. To illustrate the application of multidimensional force spectra, part two considers the design example of an S1-273 CNC lathe. First the force distribution of transmission elements is calculated. Then the equivalent fatigue force with specified reliability for the important transmission elements of the lathe under various working conditions is calculated. Use of the equivalent fatigue force determined to design the CNC lathe element enables obtaining the expected reliability.The traditional engineering approaches are to design safety margins, or safety factors, into the equipment. The safety factor (SF) is defined as the ratio of the capability of the system to the force placed on the system. The safety margin (SM) is the difference between the system capability and the force. Failure will occur if the safety factor is less than 1 or the safety margin becomes negative. This is often a deterministic approach that ignores the variability present in both the forces placed on a system and the system’s ability to react the force By the late 1930s both forces and strengths were being commonly expressed as statistical distributions. The probability and statistics theories were employed into engineering design To a large degree, reliability is an inherent attribute of a system, component, or product. As such, it is an important consideration in the engineering design process. Reliability design is an iterative process that begins with the specification of reliability goals consistent with cost and performance objectives. Once the reliability goals have been established, these goals must be translated into individual component, subcomponent, and part specifications The S1-273 lathe studied in this paper is an all-functioned CNC lathe fabricated by a Chinese machine-building mill. A schematic diagram of the main transmission is shown in . The lathe is driven by a continuous speed regulation alternating current (AC) motor by means of a timing belt and two pairs of shifting slide gears. The timing belt transmits the power and motion from the motor to shaft I in the headstock. The diameters of the driving and the driven pulley are 125 and 230 mm respectively. The motion through the gear pairs with teeth 28/70 and 49/49 is transmitted from shaft I to shaft II, and through the gear pairs with teeth 59/47 and 30/76 from shaft II to shaft III, i.e. the spindle.The rated speed of the motor is 1500 rev/min, its maximum speed is 3000 rev/min and rated power is 11 kW. The rated speed is defined as the speed at which the motor may output the maximum torque or power, in general, it is the lowest speed that can deliver the rated power The speed diagram of the lathe is shown in . The vertical coordinate is the speed in logarithmic scale. The transmission shafts are shown as vertical parallel lines at equal distances from each other. The Roman numerals at the bottom of the diagram correspond to the numbers in . The points on each vertical line indicate the margins of speed ranges for each speed level. The lines jointing the speeds between two axes indicate the transmission ratios. The numbers beside the oblique (or horizontal) lines are the numbers of teeth of the meshing gears, corresponding to . Equal gear ratios are represented by parallel lines.The speed diagram indicates directly not only the actual speed ranges of the various shafts and gears, but also the intermediate gear pairs through which these speeds are obtained. For example, shaft I has one level of speed range from 125 to 1630 rev/min obtained from the motor through the pulley pair 125/230; the spindle speed range from 20 to 250 rev/min is obtained from the motor through the pulley pair 125/230, the gear pair 28/70 between shafts I and II and the gear pair 30/76 between shafts II and III.The spindle speed range is divided into four speed levels, i.e., level M21: 20–250 rev/min, level M22: 250–630 rev/min, level M23: 315–800 rev/min and level M24: 800–2000 rev/min. M21, M22, M23 and M24 are miscellaneous codes of NC program and instruct each of the four speed levels respectively. It should be noted that level M22 and M23 overlap within the speeds of 315–630 rev/min.The main transmission expression is given asFor four speed levels mentioned above, their individual transmission ratios and corresponding spindle speeds are given in . ns is the spindle speed (in rev/min), nm is the motor speed (in rev/min), which changes continuously from 0 to 3000 rev/min, u1, u2, u3, u4 are the transmission ratios of each level respectively, nj is the basic speed of spindle for each level corresponding to the rated speed of the motor.Take the gear with 28 teeth mounted on shaft I as an example. From , it can be seen that there are two routes from shaft I to shaft III while the 28-tooth gear on shaft I is working. One is that the motion is obtained from the motor through the pulley pair 125/230, the gear pair 28/70 between shafts I and II and the gear pair 30/76 between shafts II and III. The corresponding spindle speeds range from 20 to 250 rev/min, relative speeds are from 0.01 to 0.125. The transmission ratio from shaft I to shaft III is u1′=0.158. The relative torque of spindle is x=TS/TR, TS is the torque of the spindle, TR is the rated torque of the spindle. The torque of the gear with 28 teeth is TZ=u1′·TS, The relative torque of the gear with 28 teeth corresponding to the rated torque of the spindle is z=TZ/TR=u1′·TS/TR=u1′·x.Another is that the motion is obtained from the motor through the pulley pair 125/230, the gear pair 28/70 between shafts I and II and the gear pair 59/47 between shafts II and III. The corresponding spindle speeds range from 63 to 800 rev/min, speeds from 315 to 800 rev/min are used in practical, the relative speeds are from 0.1575 to 0.4. It should be noted that the speeds from 315 to 630 overlap with those of another speed level; the relative speeds are from 0.1575 to 0.315. The transmission ratio from shaft I to shaft III is u2′=0.502. The torque of the gear with 28 teeth is TZ=u2′·TS, The relative torque of the gear with 28 teeth corresponding to the rated torque of the spindle is z=TZ/TR=u2′·TS/TR=u2′·x. To simplify the calculation, we assume that the chances of occurrences of these two speed levels are equal, that is, the probability of occurrence of each speed level is 0.5 within the speeds from 315 to 630 rev/min.Thus, the probability density function of force distribution for the gear with 28 teeth mounted on shaft I is,where, f(x,y) is the multidimensional force spectrum of the medium-sized CNC lathes.α is the normalized factor, that satisfies the following equation,The probability density function of force distribution for the gear with 28 teeth mounted on shaft I is shown in By the similar way, the force distributions of other elements can be obtained.The fatigue strength is generally less than would be observed under a static force. Fatigue testing results in experimental data relating the number of cycles to failure (C) to the magnitude of the cyclical stress (S) or force (T). The fatigue strength is the maximum stress amplitude for a specified number of cycles until failure. Mathematically, an S–C curve may take the following form where m>0 is constant determined experimentally in laboratory tests that duplicate the amplitude, frequency, and pattern of specific stresses. T is the force or stress of element, C is the number of cycles to failure.where, C0 is the standard number of cycles to failure, T′ is the limit force corresponding to C0, KT is the service life factor, determined by,where, n is the speed of element in rev/min; tm is the expected service life in hours.While the speed and force of element are variables, we plot the S–C curve and force spectrum in the same coordinate system, shown as . We divide the force interval into N equal segments, for a specific force Ti, the cumulative number of cycles in the area of Ti to Ti+ΔT, can be expressed into Ci·ΔT. Based on the Miner hypothesis of cumulative damage, the following equation can be obtained,where, Ci is the number of cycles corresponding to Ti, TR is the rated force of the element, KS is called the equivalent force factor, thus,where, CΣ is the sum of number of cycles, CΣ=60·E[n]·t; t is the expected service life (in hour); E[n] is the expectation of the speed ( in rev/min).where, z=Ti/TR is the relative force of the element; f(z) is the probability density function of relative force of the element, KP and Kn are called power utilization factor and speed change factor respectively.For the gear with 28 teeth mounted on shaft I, substituting the probability density function of relative force f(z) into The expectation of the speed of the gear with 28 teeth can be calculated by,where, f(y) is the speed spectrum of the CNC lathe; nmax is the maximum speed of the CNC lathe in rev/min.Substituting u1=0.158, u2=0.502 and nmax=2000 r/min into , we obtain E[n]=506 r/min. Suppose t=20,000 hours, m=6.6 and C0=107, thus,Thus the equivalent force factor of the gear with 28 teeth is,The above-mentioned standard number of cycles, C0 is obtained under the assumption of deterministic fatigue life, i.e., with the reliability R=0.5. While supposing the fatigue life as stochastic variable, where, KR is called reliability factor; C0R is the fatigue life with the specific reliability R.For example, while the fatigue life of element follows Weibull distribution, the reliability R(C) becomes where, Ca is the size parameter of Weibull distribution; b is the shape parameter of Weibull distribution.C0R can be calculated by the following equation Thus, the equivalent fatigue force factor with the specified reliability becomes,Furthermore, the equivalent fatigue force with the specified reliability can be calculated by the following equation,For the S1-273 CNC lathe, the rated force is,where, η is the mechanical efficient factor, in the example, η=0.88, nB=125 rev/min is the basic speed of the spindle, P=11 kW is the rated power of the lathe.After force spectrum of the element is determined, if the distribution of fatigue life of the material used is known, the equivalent fatigue force factor and equivalent fatigue force can be calculated through the above-mentioned equations.To a large degree, reliability is an inherent attribute of a CNC machine tool. As such, it is an important consideration in the engineering design process. Reliability design is an iterative process that begins with the specification of reliability goals consistent with cost and performance objectives. Once the reliability goals have been established, these goals must be translated into individual component, subcomponent, and part specifications. After individual component and part requirements have been determined, various design methods can be applied in order to meet the goals.CNC machine tools feature high speed, high efficiency, high accuracy and high automation. They must not only be extremely accurate and highly automated, but also work reliably for long periods of service. Only by means of the modern design theory and reliability method can CNC machine tools work with desired reliability.To carry out reliability design, the distribution of forces acting on the element must be determined first. The determination of force spectra and establishment of force spectra database lay a foundation for implementing reliability design for CNC machine tools.In this paper, a linear theory for the analysis of beams based on the micropolar continuum mechanics is developed. Power series expansions for the axial displacement and micro-rotation fields are assumed. The governing equations are derived by integrating the momentum and moment of momentum equations in the micropolar continuum theory. Body couples and couple stresses can be supported in this theory. After some simplifications, this theory can be reduced to the well-known Timoshenko and Euler–Bernoulli beam theories. The nature of flexural and longitudinal waves in the infinite length micropolar beam has been investigated. This theory predicts the existence of micro-rotational waves which are not present in any of the known beam theories based on the classical continuum mechanics. Also, the deformation of a cantilever beam with transverse concentrated tip loading has been studied. The pattern of deflection of the beam is similar to the classical beam theories, but couple stress and micro-rotation show an oscillatory behavior along the beam for various loadings.In the classical continuum mechanics, the motion of material particles with three (translational) degrees of freedom are described by position vectors identifying the location of each particle as a function of time (). At each particle of a micropolar continuum, it is assumed that there is a micro-structure which can rotate independently from the surrounding medium (). So every particle contains six degrees of freedom, three translational motions which are assigned to the macro-element and three rotational ones which are referred to the micro-structure. In the classical continuum theory, from the kinetic point of view, the effect of a surface element on a neighboring one is expressible by a traction vector. In the micropolar theory, the interaction between two adjacent surface elements is considered via a couple vector in addition to the traction vector (). The traction vector (per unit area) constructs the stress tensor and the couple vector (per unit area) constructs the couple stress tensor. Balance of moment of momentum shows that the stress tensor in the micropolar media is no longer symmetric as in the classical continuum theory. The non-symmetric stress tensor can be decomposed into its symmetric and skew-symmetric parts. The symmetric part causes the deformation of the macro-element and its skew-symmetric part contributes to the rigid rotation of the micro-structure with respect to the material.An analytical solution for the bending of a curved beam of micropolar elastic solid was presented by . A bending analysis of micropolar elastic beams using a 3-D finite element method has been developed by . They first derived an analytical solution for straight beam problems based on the theory of material strength. Then, they applied a new 3-D finite element to solve both straight and curved beam problems. The theory of micropolar plates based on the assumption of power series expansions for the displacement and micro-rotation fields has been developed by . This theory is an extension of the Mindlin–Reissner first order shear deformation plate theory (FSDPT) to the micropolar continuum mechanics. After some simplifications, this theory can be reduced to the Mindlin–Reissner and Kirchhoff plate theories.In this work, a linear theory for analyzing the deformation of beams based on the micropolar continuum mechanics is developed. We have followed Eringen's method for constructing the micropolar plate theory (). Equations of motion are derived by integrating the momentum and moment of momentum equations in the micropolar continuum theory. Power series expansions for the axial displacement and micro-rotation fields are assumed in the development of this theory. The terms in these series are selected so that the proposed theory is actually an extension of the Timoshenko beam theory (first order shear deformation beam theory, FSDBT) in the classical continuum mechanics to the micropolar continuum theory while constant micro-rotation field through the beam height has been assumed. Based on the developed theory, the nature of flexural and longitudinal waves in the infinite length beam is investigated. Also, a cantilever beam with transverse concentrated tip loading has been studied.The theory of micropolar continuum mechanics is well-posed and the mathematical foundations have been developed by 1971 through the works of Eringen and his coworkers. In this section, we present some basic relations of the micropolar elasticity used in the next sections based on the fundamental work of Let u denote the displacement vector of a macro-element in the continuum. Angles of rotation of the associated micro-structure constitute a rotation (pseudo) vector φ. In the case of small deformation theory, the components of the infinitesimal micropolar strain tensor e and the infinitesimal wryness tensor κ can be written as where comma denotes differentiation with respect to the rectangular Cartesian coordinates xi and εijk is the permutation symbol. Also, summation convention holds for the repeated indices.In the micropolar theory, the interaction between two adjacent surface elements is considered via a traction and a couple vector. The traction vector (per unit area) constructs the asymmetric stress tensor σ and the couple vector (per unit area) constructs the couple stress tensor m. In the micropolar theory, balance of linear momentum and moment of momentum are in the following form:σji,j+ρ(fi−u¨i)=0,mji,j+εimnσmn+ρ(li−Jφ¨i)=0, where ρ is the current density, fi and u¨i=∂2ui(x,t)/∂t2 are the body force density and the acceleration component in the i-direction, J is the micro-inertia, li and φ¨i=∂2φi(x,t)/∂t2 are the body couple density and micro-rotation acceleration component in the i-direction, respectively. In the linear theory of micropolar elasticity, the equations of motion are supplemented by the following constitutive equationsσij=λekkδij+(μ¯+η)eij+μ¯eji,mij=ακkkδij+βκij+γκji, where λ=Eν/[(1−2ν)(1+ν)], μ¯=G−η/2, η, α, β and γ are elastic constants. Here, E, ν and G correspond respectively to the Young modulus, Poisson ratio and shear modulus in the classical theory of linear elasticity.Consider a beam with length L, height 2h and variable width b (along length and height of the beam) as shown in . A rectangular Cartesian coordinate system {x,y,z} with base vectors {i,j,k}, is located at one end of the beam where the coordinate x is along the length and z is along the height of the beam. Also, y-axis is coincident with the neutral axis of the beam, where the definition of the beam neutral axis is the same as the definition used in the classical beam theories. The beam cross section is assumed to be symmetric with respect to the z-axis. The geometry of the beam in the coordinate system can be described with 0⩽x⩽L, b=b(x,z) and −h−⩽z⩽h+. We will construct the micropolar beam theory based on the following assumptions:– The beam height 2h and its width b(x,z) are small as compared to the length L.– The elastic constants are functions of x-coordinate.– Distributions of the body force, body couple, surface traction, surface couple, strain, wryness, stress and couple stress components are independent of y-coordinate.– The stress and displacement fields do not vary severely across the height.– The loading is so that no torsion occurs in the beam.– The non-zero components of stress, couple stress, body force and body couples are σxx, σxz, σzx, mxy, mzy, fx, fz and ly. However, in the next section we show that mzy is also zero in this theory.Using the above assumptions, the simplified forms of the equations of motion ∂σxx∂x+∂σzx∂z+ρ(fx−u¨x)=0,∂σxz∂x+∂σzz∂z+ρ(fz−u¨z)=0, where ux, uz and φy are displacement components along the x- and z-axes and the micro-rotation about the y-axis, respectively. Also, ρ and J are functions of x-coordinate. The upper and lower faces of the beam are subjected to the traction and couple vectors ttop(k), tbot(−k), mtop(k) and mbot(−k), respectively. The resultant traction τ and couple vector μ take the following formτ=ttop(k)−tbot(−k)=[τtop−τbot]i+[ptop−pbot]k=τi+pk, It is noted that τ, p and m are quantities per unit length of the beam.There are two well-known methods for deriving the “approximate” theories of beam, plate and shell structures. The first one is the “Asymptotic Expansion Method” and the second one is the “Power Series Expansion Method”. Although the first step of the asymptotic approach is also a power series expansion of different parameters, but the next steps are completely different with that of a direct “Power Series Expansion Method”. has used the power series expansion method for deriving the equations of his first order micropolar plate theory, while have used the asymptotic approach. As it is shown in the latter paper, the equations of motion obtained by these two derivation methods differ essentially with each other. In other words, the results of these two methods are not the same and they may be regarded as two completely different approximative approaches. In the context of the classical continuum theory, the power series method has been frequently used in the literature. Here, we only refer to and references therein. Also, asymptotic approach has been applied frequently for deriving the equations of beam, plate and shell theories in the classical continuum mechanics. Among them, we refer to the well-known classical book by . For the classification of the models we refer to . Also, for the case of the “geometrically exact” theories of rods, plates and shells, we refer to the pioneer work by . Recently, a geometrically exact theory for large deformation analysis of the micropolar shells has been developed by In this paper, we are interested to the power series expansion method which is frequently used in the engineering literature. However, construction of a micropolar beam theory based on the asymptotic method can be performed by an alternative formulation which is beyond the scope of this work.According to the basic assumptions mentioned in the previous section, we assume power series expansions for the displacement component ux and the micro-rotation field φy in the following formux(x,z,t)≈∑i=0∞ziψ(i)(x,t),φy(x,z,t)≈∑i=0∞ziφ(i)(x,t). Also, similar to the classical theories, we do not consider any power series expansion for the displacement component uz and employ the following simplest approximation In general, one may construct a (r,s) approximate theory by considering the following approximate fields for ux and φyux(x,z,t)≈∑i=0r⩾0ziψ(i)(x,t),φy(x,z,t)≈∑i=0s⩾0ziφ(i)(x,t). we get the following relation for strain components in terms of the non-symmetric stress componentseij=1η(η+2μ¯)[(μ¯+η)σij−μ¯σji−λη3λ+2μ¯+ησkkδij],(i,j,k=1,2,3). By setting μ¯=G−η/2,σij=σji and letting η tend to zero, Eq. reduces to the well-known generalized Hooke's law for isotropic elastic materials in the classical continuum mechanics. Using the assumption σyy=σzz=0 in Eq. , exx, eyy and ezz have the following relations with σxxexx=2λ+2μ¯+η(2μ¯+η)(3λ+2μ¯+η)σxx,eyy=ezz=−λ(η+2μ¯)(3λ+2μ¯+η)σxx=−νexx. It is important to note that from the assumed displacement field in Eq. one may write ezz=∂uz/∂z=0, but similar to the classical beam theories, we use the value of ezz=−νexx obtained from the constitutive equation as given in Eq. 1 we obtain exx=σxx/E which is well known in the classical continuum beam theories. In an analogy with the classical beam theories, Eq. may be written as exx=σxx/E¯, where E¯ is defined in the following form From the assumed displacements and micro-rotation fields given in Eqs. , the micropolar strain components exx, exz and ezx and their corresponding stress components are as followsexx=∂ux∂x=∑i=1rzi∂ψ(i)∂x,ezx=∂uz∂x+φy=∂w∂x+∑i=0sziφ(i),exz=∂ux∂z−φy=∑i=1r⩾1izi−1ψ(i)−∑j=0szjφ(j),σxx=E¯∑i=0rzi∂ψ(i)∂x,σzx=(μ¯+η)∂w∂x+μ¯∑i=1r⩾1izi−1ψ(i)+η∑j=0szjφ(j),σxz=μ¯∂w∂x+(μ¯+η)∑i=1r⩾1izi−1ψ(i)−η∑j=0szjφ(j).In what follows, we construct a simple micropolar beam theory by choosing some special terms in the general power series expansions given in Eqs. , by setting r=1 and s=0 we obtain a (1,0) theory with the following approximate displacement and micro-rotation fieldsux(x,z,t)≈∑i=0r=1ziψ(i)(x,t)=u(x,t)+zψ(x,t),φy(x,z,t)≈∑i=0s=0ziφ(i)(x,t)=ϕ(x,t),uz(x,z,t)≈w(x,t), where u, ψ and ϕ have been used for ψ(0),ψ(1) and φ(0), respectively. This is similar to the Timoshenko's First Order Shear Deformation Beam Theory (FSDBT) in the classical continuum theory with additional constant micro-rotation field through the beam height. on the beam cross section with the aid of Eqs. and dividing the results by the cross sectional area A(x) we obtain In the above equations, the following definitions have been used{Nxx,Nxz,Nzx,Mxy,f¯x,f¯z,Ly,u¯x,u¯z,φ¯y}=1A∫∫A{σxx,σxz,σzx,mxy,fx,fz,ly,ux,uz,φy}dA.1 by z and integrate on the beam cross section which yields the following equation where M, f˜x, u˜x and τ¯ are defined as{M,f˜x,u˜x}=∫∫A{σxx,fx,ux}zdA,τ¯=1h(h+τtop−h−τbot). are the balance equations for the (1,0) micropolar beam theory. From Eqs. where I is moment of area of the beam cross section about y-axis. Also, the wryness component κxy and the average couple stress resultant Mxy have the following form As it was mentioned in the previous section, in this theory the couple stress component mzy vanishes, since ∂ϕ(x,t)/∂z=0. Also, the value of σxx has the following relation with Nxx and M Furthermore, the total amount of bending moment at the beam cross section, M¯, can be obtained from the following equation It is important to note that from Eq. () for the case of (m=1,n=0), this theory predicts a constant distribution of transverse shear σxz across the height of the beam which is similar to the Timoshenko beam and Mindlin–Reissner plate theories. In the classical beam theories, unless a shear traction is applied to the top or bottom surfaces, transverse shear must vanish at these surfaces. It is well known that for the beams with rectangular cross section, the distribution of the transverse shear stress is quadratic through the height of the beam and vanishes at top and bottom surfaces. Therefore, a constant shear distribution overestimates the shear energy. Thus, a correction factor is often used to reduce the energy associated with the transverse shear. The inconsistency of the Euler–Bernoulli beam or Kirchhoff plate theory is even more severe, since the kinematic assumption results in a vanishing transverse shear and hence, these theories are inconsistent with equilibrium. In the micropolar media, σxz≠σzx and hence, there is no inconsistency with equilibrium equations. Neglecting the micropolar effects, the micropolar beam theory should reduce to the classical theories. In order to have a quadratic distribution of σxz through the height of the beam, a third order theory by setting m=3 in Eq. 1 similar to the classical third order beam theory may be constructed. We may improve the micropolar beam theory by choosing more terms in the power series expansions presented in Eqs. . In fact, a proper micropolar beam theory should yield the following relation for σxzσxz=1b∂∂x(MQI)+σxzτ+σxzφ=VQbI+MbI∂Q∂x−MQbI2dIdx+σxzτ+σxzφ, where V is the shear force in the beam cross section, Q=∫zh+zdA is the first moment of area of the beam cross section, σxzτ is the effect of shear loading on the lower and upper surfaces of the beam and σxzφ is the part of the transverse shear σxz resulting from the micro-rotation (pseudo) vector φ. It is noted that the first four terms of Eq. are present in the theory of elasticity in the classical continuum mechanics (), while the last term is specific to the micropolar media.Substituting the stress and couple stress resultants given in Eqs. with the aid of μ¯=G−η/2, we obtain the following governing equations of motion for the micropolar beam in the (1,0) theory∂∂x{A[(G+η2)ψ+(G−η2)∂w∂x−ηϕ]}+p+ρA(f¯z−w¨)=0,∂∂x(Aβ∂ϕ∂x)+ηA[∂w∂x−ψ+2ϕ]+m+ρA(Ly−Jϕ¨)=0,∂∂x(E¯I∂ψ∂x)−A[ηϕ+(G+η2)∂w∂x+(G−η2)ψ]+hτ¯+ρ(f˜x−Iψ¨)=0. It is noted that variational formulation can also be used as an alternative for obtaining the governing equations of motion. The axial extensional motion u(x,t) in Eq. is decoupled from the other components of motion and is identical with the axial extensional motion in the classical beam theories with the exception of replacing E¯ for E. By setting η=0 in Eqs. we will obtain the equations of the Timoshenko beam theory (without shear correction factor) in the classical continuum theory. By neglecting transverse shear effect (G→∞) in the Timoshenko beam theory, one can obtain the Euler–Bernoulli beam theory.In the case of static problems and constant material and geometric properties, state space method can be used for solving differential equations . As it was mentioned in the previous section, Eq. is a decoupled differential equation in terms of only u and can be solved independent of other equations. Now we introduce the following state variables we obtain the following system of the first order differential equationsX1′=X2,X2′=1E¯I{A[ηX3+(G+η2)X5+(G−η2)X1]−hτ¯−ρf˜x}, where the prime stands for differentiation with respect to x-coordinate. The system of equations in may be written in the following form X′=AX+B with X={X1,X2,…,X5}t andA=[01000A(G−η/2)E¯I0AηE¯I0A(G+η/2)E¯I00010ηγ0−2ηγ0−ηγ0−G+η/2G−η/20ηG−η/20],B={0,−hτ¯+ρf˜xE¯I,0,−1β(mA+ρLy),−1G−η/2(pA+ρf¯z)}t.λi=0,0,0,±iξwithξ=ηG(E¯I+Aβ)E¯Iβ(G−η/2). And the solution for the vector X is in the following formX(x)=[I+xA+x22A2+ξx−sinξxξ3A3+1ξ4(cosξx−1+12ξ2x2)A4]X(0)+[xI+x22A+x36A2+(x22ξ2−1−cosξxξ4)A3+(sinξxξ5−xξ4+x36ξ2)A4]B, where X(0) is the value of X at x=0 and are determined by the boundary conditions of the problem.In this section we examine the nature of flexural and longitudinal waves in an infinite length micropolar beam. At first we define functions Ψ and Φ as follows{w(x,t),Ψ(x,t),Φ(x,t)}t={W0,Ψ0,Φ0}texp[i(ωt+kx)], where ω is the natural frequency of the wave, k=2π/λ¯ is the wave number with λ¯ as the wave length and W0,Φ0 and Ψ0 are constants. Substituting Eqs. where the coefficient matrix H has the following formH=[(G−η2)k2−ρω2(G+η2)k2−ηk2−A(G+η2)ρIω2−A(G+η2)−E¯Ik2−Aηη−ηη−γk2+ρJω2]. For existence of a non-trivial solution (W0,Φ0,Ψ0), it is necessary for the determinant of the coefficient matrix H to vanish. This leads to an algebraic equation of six degree for the wave speed C=ω/k. If we neglect rotary inertia effect (ρI→0), we get the following biquadratic equation where using Λ=2hk, the following definitions have been usedα2=E¯Iρ[J(G−η/2)+β]4Ah(G+η/2)Λ4−ρh[A(G+η/2)(Jη−β)+E¯Iη]A(G+η/2)Λ2−4ρηh3(G+2η/3)G+η/2,α3=E¯Iβ(G−η/2)2Ah(G+η/2)Λ4−2hη(E¯I+Aβ)AΛ2−24h3η2(G+η/6)G+η/2. which shows that the waves corresponding to the above wave velocities are dispersive, since they depend on the wave frequencies. When the micro-inertia J vanishes, we have α1=0 and hence C2(Λ)=α3/(2α2). Assuming that J, or equivalently α1α3/α22 is small, by using the first order Taylor series approximation 1−α1α3/α22≈1−α1α3/2α22, we can approximate Eqs. The flexural wave speeds C1(Λ) and C2(Λ) can be plotted against the non-dimensional parameter Λ. For the case in which h=0.1m, b=0.05m, E=20GPa, ν=0.3, η=G/20, β=G/5000 (magnitudely), ρ=2000kg/m3 and J=5×10−5m2, the non-dimensional wave speeds C1/Cs and C2/Cs versus Λ are shown in . Here Cs=(G+η/2)/ρ corresponds to the speed of equivoluminal elastic wave. For the values of Λ smaller than a specific value, the value of C1(Λ) is imaginary.By setting C1(Λ)=0 we find that this value of Λ is a root of the algebraic equation α3(Λ)=0. For the given data, this value is Λ∗=3.3758. C1(Λ) and C2(Λ) have the following limitsC1(Λ∗)=0,limΛ→∞C1=G−η/2ρ,limΛ→0C2=3η(G+η/6)ρ(G+2η/3),limΛ→∞C2=βρJ. It is noted that for τ=fx=0 and u=U0exp[i(ωt+kx)], Eq. yields to C=E¯/ρ which is the non-dispersive speed of longitudinal wave along the beam. It is very similar to the classical theory, but the Young's modulus Eis replaced by E¯.Consider a beam with following geometric and material properties: L=1m, h=0.1m, b=0.05m, E=20GPa, ν=0.3, η=G/20, β=G/5000 (magnitudely) and ρ=2000kg/m3. Let the beam be fixed at one end and subjected to a transverse concentrated load Fin the positive z-direction at the other end. Other loading parameters are Ly=(∫∫AlydA)/A and m=mtop−mbot distributed along the beam as shown in we obtain u(x)=0. The boundary conditions for the micropolar beam are as follows@x=L:{b∫−hhτxzdz=F⇒2bh[μ(X1+X6)+η(X1−X3)]=F,M=mxy=0⇒X2=X4=0. have obtained the following formula for the tip deflection of a cantilever micropolar beam under the transverse concentrated tip load F where l=2β/E is a length-scale parameter. It is noted that the micropolar length-scale parameter is defined in various forms in the literature. Some authors define l¯=β/(4G) (see e.g. () as the micropolar length-scale parameters. For the given material parameters, we will have l=0.0124≪1 and Eq. yields the non-dimensional tip deflection 3EIw(L)/FL3=0.9774. The proposed micropolar beam theory, for the case of m=Ly=0, predicts the value of 3EIw(L)/FL3=0.9810 which is very close to the result of the theory developed by . The value of the non-dimensional tip deflection 3EIw(L)/FL3 for different theories and different loadings of m and Ly is summarized in The variation of the non-dimensional normalized deflection 3EIw(x)/FL3 along the beam for different values of m and Ly is shown in . From this figure, it is clear that the developed theory, for the case of m=Ly=0, predicts a deflection pattern slightly smaller than the classical Euler–Bernoulli beam theory.The variation of the couple stress component mxy (for the case of F=1000N) along the beam for different values of m and Ly is shown in . The oscillatory behavior is due the predominance of the effect of the eigenvalues ±iξ in Eq. where the frequency of the oscillations is equal to ξ/2π.The variation of the micro-rotation ϕ (for the case of F=1000N) along the beam for different values of m and Ly is shown in . Again, the effect of the eigenvalues ±iξ exhibits a slightly oscillatory behavior in the pattern of ϕ, but this effect is less as compared to the diagrams of the couple stress component mxy.In this paper, a linear theory for the deformation of beams based on the micropolar continuum mechanics has been developed. Power series expansions for the axial displacement and micro-rotation fields are assumed for the development of this theory. Equations of motion governing in the micropolar beams are derived by integrating the momentum and moment of momentum equations in the micropolar continuum theory. Body couples and couple stresses can be supported in this theory. After some simplifications, this theory can be reduced to the well-known Timoshenko and Euler–Bernoulli beam theories. This theory predicts a constant distribution of transverse shear σxz across the height of the beam which is similar to the Timoshenko beam and Mindlin–Reissner plate theories. We may improve the micropolar beam theory by choosing more terms in the power series expansions of displacements and micro-rotation fields. The nature of flexural and longitudinal waves in the infinite length micropolar beam has been investigated. This theory predicts the existence of micro-rotational waves which are not present in any of the known beam theories based on the classical continuum mechanics. Moreover, all types of the flexural waves are dispersive, since they are dependent on the wave frequencies. The longitudinal wave predicted by this theory is not dispersive and is very similar to the results of the classical beam theories. Also, the deformation of a cantilever beam with transverse concentrated tip loading has been studied. The pattern of deflection of the beam is similar to the classical beam theories, but couple stress and micro-rotation show an oscillatory behavior along the beam for various loadings.Some aspects of damage and failure mechanisms at high strain-rate and elevated temperatures of particulate magnesium matrix compositesIn this paper, a study of the damage and fracture mechanisms on a particulate metal matrix composite of magnesium base alloy, reinforced with SiC particles, is presented. Quasi-static and dynamic tensile tests were carried out at room and elevated temperatures. The effects of the temperature and strain-rate on the broken and decohered particles were determined in the fracture zone and along the gauge length of the tested specimens. In quasi-static tests at room temperature the percentages of broken and decohered particles are similar; a temperature rise increases the percentage of decohered particles much more than the broken ones which maintains nearly constant. For dynamic tests, the percentage of broken particles is greater than of the decohered ones; the influence of the temperature on the percentage of broken or decohered particles is much less than in the quasi-static tests.Recently, there has been a growing interest in the use of Metal Matrix Composites (MMCs) for high strength structural applications such as lightweight, etc. In particulate MMCs, ceramic particles (alumina, silicon carbide, silicon nitride, etc.) are incorporated in metallic alloys to improve their mechanical behaviour The mechanical properties of the MMCs are related to their work conditions and microstructure, which could be governed by the manufacturing process. In the course of deformation of the particulate MMCs, particles are broken or separated from the surrounding matrix (decohered) by the load transmitted by the matrix. The resulting microvoids are the origin of the growth and coalescence processes in the failure and damage mechanisms The influence of these damage and failure mechanisms, individually and collectively, on the overall deformation and fracture resistance is strongly dictated by factors as diverse as: (i) the size, shape and spatial distribution of the reinforcement, (ii) the concentration of impurities in the constituent phases of the composites, (iii) the processing and heat treatment procedures, including ageing treatments, to which the composite is subjected prior to mechanical loading, and (iv) the thermal and chemical environment, in service conditions The first stage of the damage process is usually void nucleation. The study of the plastic deformation and fracture processes The initiation of microcracks can be greatly affected by the characteristics of the second-phase particles; a common mechanism of void nucleation at second-phase particles occurs by the separation of particles from the matrix at the particle–matrix interface, a process known as interface decohesion.The second stage of damage mechanism is the growth and coalescence through the metallic matrix of the microvoids nucleated at the reinforcements. The progress of a microvoid is attributed mainly to the plastic deformation of the material around the void. The void growth is controlled by the strain produced by the load conditions.Several studies have been made of the mechanical behaviour of MMCs, and their damage and fracture, mainly under quasi-static loading. Most of these studies consist of testing specimens of the materials and observing them under the Scanning Electronic Microscope (SEM) to detect the damage characteristics in the matrix and in the reinforced particles.The damaged proportion of particles (broken or decohered) gives the input for the numerical models that can predict and evaluate the mechanical behaviour of the MMCs during modelling of particle cracking with different forms In a wide range of engineering applications, these materials may be subjected to extreme loading conditions at high strain-rates. The mechanical characteristics of pure metals and alloys have been reported in classical studies The service temperature is also a determining factor in the mechanical behaviour of materials. Some studies have investigated the influence of this parameter on the mechanical behaviour of MMCs such as the deformation, damage and fracture mechanisms Magnesium alloys are attractive industrially as a result of their low densities. During the last decade the main markets for these alloys, although with different trends, were the aeronautical This study focuses on the phenomenological description of the final failure mechanisms in dynamic tensile tests at different temperatures, using a commercial magnesium alloy reinforced with silicon carbide particles. The results are presented and discussed in relation to the damage produced in the ceramic particles as revealed by observation under the microscope.The investigated materials in this study were a commercial extruded magnesium alloy composed of 94 wt% Mg, 5 wt% Zn and 1 wt% Mn and the same alloy reinforced with 12 vol% SiC particles. The average particle size was 10 μm. The materials were manufactured by extrusion. The row material and the composite were heat treated by the manufacturer to T6 state (solution at 435 °C for 8 h, quenched, and then aged at 200 °C for 16 h). The materials were supplied by Magnesium Electron, MELRAM composite, England.A microstructural analysis of the composite was carried out It was noted from the microscopic observation that the reinforced particles have irregular shapes that can be approximated by cylinder shapes with the major axis orientated in the extrusion direction. The longitudinal section of a generic particle can be approximated by a rectangle of sides a and b. The particle size is defined as the square root of the area and the aspect ratio as a/b (where a>b). The distribution of size and the aspect ratio of particles are shown in . The average particle size obtained from this distribution is 11.3 μm, and the average aspect ratio is 2.0.The population of particles was divided into three separate groups, marked off by a 10% frequency in the frequency-size curve, where the separating values of the three intervals are d0=7.5 μm and d1=16 μm. A particle size smaller than or equal to do is classified as ‘small’, those between d0 and d1 are classified as ‘medium’, and those larger than d1 as ‘large’. The average values of the three classified particle-sizes are shown in Regarding the distribution of the reinforced particles in the matrix, it was observed that in longitudinal sections, the particles were distributed in two bands: dense-bands and no-dense-bands. In the dense-bands, the proportion of particles is higher than in the no-dense-bands, as shown in . The distribution of particles in the matrix, in both bands, is observed to be almost uniform. The micrographs of this magnesium composite showed no clear evidence of clusters of particles in the matrix, although some agglomeration of particles was observed, such as that shown in , which accumulates porosity in the matrix. This is because the small interparticle distance hinders the matrix penetration among the reinforced particles, and consequently produces more porosity in this zone.The distributions of ceramic particles in the different zones, taken from the micrographs, are shown in . The percentages of particles in each type of the defined classes (‘small’, ‘medium’ and ‘large’), with respect to the total number of particles, are almost similar in the dense and light zones.Quasi-static tests were performed on a universal testing machine (Instron, model 8516) of 100 kN maximum load capacity. Cylindrical standard tensile specimens of 25.4 mm gauge length and 6.35 mm diameter were used (. Specimens were machined according to standard specification ASTM E-8 A Split Hopkinson Pressure Bar device was used for the dynamic test at a strain-rate of 600 s−1. Dynamic tests were carried out following the ESIS (European Structural Integrity Society) recommendations The specimens were tested at the following temperatures: room temperature; a temperature just below the ageing temperature (200 °C); and 100 °C as an intermediate temperature to establish the effect of temperature on the mechanical behaviour. A furnace coupled to each testing equipment were used to obtain the required temperatures of 100 °C and 200 °C. Two periods of heating times were adopted: 10 min as an average time for attaining the temperature throughout the specimen, and 2 h which may produce microstructural effects The evolution of damage was determined on each tested specimen by microstructural analyses, observing the reinforced material with a Philips XL30 Scanning Electronic Microscope (SEM) with a maximum acceleration voltage of 30 kV.The two parts of the tested broken specimen were used; one for the observation of the fracture surface and the other was cut in a longitudinal section (parallel to the extrusion direction) conserving one its ends to be the fracture surface. Cutting operations were carried out with a diamond thread cutter.Grinding and polishing processes were carried out as usual in the literatures. The only difference was in the finishing process, at the beginning of the this study, pure methanol was used as a cooling medium but it reacts with the magnesium alloy and the composite material. So the pure methanol was avoided in the finishing process. Finally, pure ethanol was used as a cooling medium which did not react with the studied materials.The results of the fractographic analysis of failed tensile specimens under quasi-static and dynamic conditions may be summarized as follows:The presence of the reinforcement improves the strength characteristics of the base alloy and reduces its ductility in quasi-static and dynamic tests, It can be observed from the true-stress true-strain curves of the base alloy and the composite material as shown in . In the reinforced material, the plastic deformations are small (the strain corresponding to the ultimate tensile strength is about 3%) as shown in and there is no appreciable necking. However, in the base alloy, plastic deformations may reach 11% in the quasi-static test and 6% in dynamic tests, and necking is observed. Furthermore, the temperature rise increases the strain values in both quasi-static and dynamic tests as revealed from the true-stress true-strain curves (The small amount of decohered particles indicates a high interfacial bonding ceramic/matrix strength, and consequently the particles withstand most of the loading transferred by the matrix, which is attributed to their high strength. This can explain the strain hardening caused in this composite material. In addition, the hydrostatic pressure in the matrix due to the existence of particles increases the stress required to produce plastic deformation.The photographs show local micromechanisms of deformation in the form of broken or decohered particles ( and also some population of cracks propagates in the matrix. These local micromechanisms were observed in the zones close to the fracture surface, decreasing with the further distance from this surface.The microvoids that appear as a result of the fracture and decohesion of particles originate growth and coalescence processes which produce microcracks in the matrix along planes perpendicular to the loading direction (see Although the failure mechanism of the reinforced material is based on the fracture and decohesion of ceramic particles and the subsequent growth and coalescence of the corresponding microvoids, the recorded low fracture strain values demonstrate that the reinforced material has a lower ductility than that of the base alloy.Macroscopic observation of the fractured specimen shows a small necking zone in both quasi-static and dynamic tests. The microscopic observation of the fracture surface of this composite has characteristic features of the void nucleation mechanism, as shown in Damage and failure in MMCs is generally associated with the breaking and decohering of the ceramic particles. The fracture and decohesion of the reinforced particles has a detrimental effect on the overall load-bearing capacity of the composite. It reduces the flow stress, strain hardening exponent, and ductility of the composite during monotonic tensile loading (horizontal direction in the photographs).Particles during the tensile test were broken as a result of the load transfer from the matrix to particles whereas decohesion of particles was related to the interface failure. When fracture or decohesion of particle occurs, it liberates its load to the surrounding matrix and the resulting higher stresses are transmitted to the unbroken particles. The duration and stability of this process depend on the strain hardening of the matrix, which is low in this case.Whatever the temperature and strain-rate, particles are broken along planes perpendicular to the loading direction. Failure occurs in almost well-defined planes, producing the separation of the parts. shows broken particles of different aspect ratios in (a) quasi-static and (b) dynamic test conditions.Quantitative microscopy analyses demonstrate that large particles were more exposed to failure than small ones; the larger the particle the greater its probability to have defects, and consequently the lower the load it can withstand before fracture. At the beginning of the test when the plastic strain is small, the large particles are the first to break. As this strain increases, medium-sized particles start to break, indicating the localisation of the fracture surface. When the fracture of the specimen occurs in quasi-static and dynamic tests, approximately 60% of the large particles, 35% of the medium particles, and 20% of the small particles are broken; these proportions were maintained at any temperature.It was noted that the probability of particle fracture increases with its aspect ratio. The higher the aspect ratio, the greater the load transferred to the particle by the matrix. Particles with a low aspect ratio have a better ‘defence’ against this transferred load. Sometimes particles were crushed, especially those with a lower aspect ratio (The study of the percentage of the broken and decohered particles (with respect to the total particles) in the near fracture zone of the specimen in quasi-static and dynamic tests as a function of the temperature conditions shows that at any strain-rate, the proportion of broken particles decreases while the proportion of decohered particles increases.In quasi-static tests, the matrix suffers a softening process due to the temperature rise, which produces a higher strain in the material and consequently a higher proportion of decohered particles. So the proportions of decohered particles increases from 12% at RT to 42% at 200 °C (. On the other hand, the higher the temperature, the smaller the capacity of the matrix to transmit the load to the reinforced particles, so, the number of broken particles decreases from 17% at RT to 12% at 200 °C.In dynamic tests, damage process is mainly controlled by the percentage of broken particles which is more than the decohered one. The proportion of decohered particles is much lower, however, than that in the quasi-static tests at any temperature; this is because the test is run at a higher strain-rate but the strain values are lower. On the other hand, the inertial forces acting on the particles increase the probability of their breaking which could explain the higher proportion of broken particles. The study of the number of broken and decohered particles with the temperature demonstrates, however, the tendency is similar to quasi-static tests (). So the proportion of decohered particles increases from 2% at RT to 5% at 200 °C and the proportions of broken particles decreases from 25% at RT to 17% at 200 °C.In quasi-static tests, the proportion of both broken and decohered particles increases from 29% at RT to 54% at 200 °C, clear evidence of the rise of deformation with the temperature. In dynamic tests, this proportion decreases from 27% at RT to 22% at 200 °C, which confirms the insignificant influence of the temperature on the mechanical behaviour.At a given temperature, a longer heating time is equivalent to an increase of the given temperature. This implies a slight decrease in the number of broken particles and a slightly higher proportion of decohered particles.A study was made of the distribution of the total number of broken or decohered particles along the gauge length of the specimen in the quasi-static and dynamic tests. shows the evolution of the broken and decohered particles as a function of their distance (in mm) from the fracture surface, in the case of the quasi-static tests (a) at room temperature (RT) and (b) at 200 °C. At RT, the distribution of the broken and decohered particles along the specimen is almost constant, with a very slight decrease of both the proportions. At 200 °C, however, the decohered particles predominate at the broken surface, decreasing notably in percentage at a zone 1 mm from the surface; the percentage of the broken particles maintains almost constant along the gauge length of the specimen.shows the evolution of the percentage of broken and decohered particles as a function of their distance (in mm) from the fracture surface, in the case of the dynamic tests (a) at room temperature (RT) and (b) at 200 °C. At RT and 200 °C, the broken particles predominate in the damage process to fracture; their proportions were reduced significantly up to 1 mm from the fracture surface maintaining almost constant along the rest of the specimen.Damage and fracture mechanisms of MMCs of magnesium base alloy were studied in quasi-static and dynamic tests at different temperatures. Microstructural analyses were carried out to evaluate the effect of the fracture of the reinforced particles.As the particles having higher aspect ratio are the first to break at any temperature and strain-rate, better mechanical properties of the composite material can be obtained by decreasing the mean value of the aspect ratio of particles. This procedure can be achieved using one method of segregation of particles.Microstructural observations show that when increasing the strain-rate, damage process till the final fracture occurs principally due to the higher percentage of broken particles (compared with the decohered ones). As a result, better mechanical properties could be obtained by decreasing the percentage of particles having high aspect ratio; which decreases the total number of broken particles.In quasi-static tests at room temperature, the percentages of broken and decohered particles are nearly similar; although, the temperature rise gives a greater increase of the percentage of decohered particles, whereas the percentage of broken particles decreases slightly. So, if these composite materials have to work at a high service temperature, particles should be treated especially to improve the particle/matrix interface, which consequently decreases the number of decohered particles.In dynamic tests at room temperature, the percentage of broken particles is higher than that of the decohered particles and the effect of temperature rise is less evident than that in quasi-static conditions.From the distributions of particles along the gauge length of the tested specimens, it is clear that in quasi-static tests at room temperature, the percentage of decohered and broken particles are very similar; whereas, at higher temperature the decohered particles increases sharply in the adjacent zone to the fracture surface. This behaviour demonstrates that although the temperature rise leads to higher percentage of the decohered particles than the broken one, this difference in percentages is still higher when reaching the corresponding strain to fracture. In the case of dynamic tests, the percentage of broken particles is always higher than the decohered one regardless the temperature or the distance from the fracture surface. However, in the adjacent zone to the fracture surface, the percentage of broken particles is more significant, which means that the final fracture is associated to the higher percentage of broken particles for high strain-rate tests.Mechanical response of polycarbonate nanocomposites to high velocity impactImpact strength, failure energy and fracture toughness were dramatically enhanced by the incorporation of a very small amount of MWCNTs into the PC matrix at low strain rates, and strain-rate sensitive at high strain rates.In this study, the mechanical responses of polycarbonate (PC) and PC/multi-walled carbon nanotubes (MWCNTs) to dynamic loadings at low and high velocities impacts were investigated experimentally using an instrumented falling weight impact tester (IFWIT) and a split Hopkinson pressure bar (SHPB), respectively. The results from the IFWIT tests revealed that impact strength, impact failure energy and fracture toughness were dramatically enhanced by the incorporation of a very small amount of the MWCNTs into the PC matrix. The maximum load and the impact failure energy increased by ∼320% and ∼350%, respectively, when only 1 wt% MWCNTs was incorporated. The results from the SHPB tests demonstrate that all the materials showed strain-rate sensitivity. The MWCNTs nanocomposites exhibited higher yield stress and energy absorption characteristics compared to the PC matrix material. However, the enhancement by MWCNTs was very limited for the PC containing higher percentage of the filler at higher strain rates. This could be resulted by a thermal-softening effect. In addition, the density of the pure PC and PC/MWCNTs nanocomposite specimens before or after SHPB testing was examined to gain insight into the microstructure changes. The results show that the density decreased significantly after the SPBH tests. With increasing strain rate the density decrease in PC nanocomposite is faster than that in the pure PC. It is believed that more cracks formed in the PC nanocomposite during the SHPB tests, which could result in high energy dissipation.As terrorist attacks have increased around the globe in recent years, the issue of combat survivability has become a primary concern for the public and governments Over the last few decades, due to its unique property combination of high mechanical strength, outstanding impact resistance, good chemical resistance and excellent transparency, polycarbonate (PC) has been widely used as a shield or anti-ballistic material in areas of military, sports and personal protection In recent years, the development of nano-reinforced composites materials has garnered great interest in the material science community. These materials, a synthesis of a base matrix polymer and particle, where at least one aspect of the particle is in the nanometre-scale, have been shown to improve a wide range of physical and engineering properties, e.g. stiffness, impact resistance, fracture toughness and the ability to absorb energy, of the matrix material Polycarbonates (PC) with medium molecular weight polycarbonate (MFI = 21.6 g/10 min) was received from SABIC Innovative Plastics. Chemically modified multi-walled carbon nanotubes (MWCNTs) with 3–5% OH attached on the side wall were obtained from Chengdu Institute of Organic Chemistry, Chinese Academy of Science. Average diameter of the MWCNTs is about 20 nm, the length about 30–50 μm. Na+ montmorillonite clay was obtained from Southern Clay Products, USA.PC and MWCNT or clay at different mass ratios was pre-mixed based on the method suggested in a patent Scanning electron microscopy (SEM) images were taken on a field emission gun SEM (FEGSEM) LEO 1530VP instrument to observe the fracture cross-sectional morphology of PC nanocomposites. Glass transition temperature of the materials was measured by using a TA instrument DSC 2920 differential calorimetry (DSC). Heating rate was 10 °C/min. Impact resistance tests were performed using the Rosand Instrumented Falling Weight Impact Tester (IFWIT). ASTM standard test method was followed to assess the impact behavior of the PC and its nanocomposites. A standard dropping mass of 10 kg was employed on the entire specimen from a drop height of 0.5 m at a speed of 3.12 m/s. five specimens for each sample were tested. The impact behavior of the PC and its nanocomposites under loading at high strain rates of 3100 s−1, 3400 s−1, 3800 s−1 and 4000 s−1 was examined using the SHPB at room temperature in the Department of Physics at Loughborough University. The details of the SHPB used in this research can be found in Ref. shows SEM images of the fracture surfaces of the pure PC and its composites with 0.5 and 1.0 wt% MWCNTs. It can be clearly seen that the nanotubes were very dispersed and embedded into the PC matrix at nanoscale level.The glass transition temperature, Tg, of the PC and its MWCNTs nanocomposites were measured by means of DSC, and the results are shown in . PC shows a glass transition temperature (Tg) at about 149 °C and the addition of nanofiller resulted in a slight decrease of Tg. for the nanocomposite systems. This phenomenon is due to the fact that the addition of nanoparticles might create more free volume in the composites, resulting in a slight decrease in Tg. relates to the performance of the PC and its MWCNTs nanocomposites under test using an instrumented falling weight impact tester (IFWIT), where the characterization involves drawing up a history of the damage kinetics of the specimens from initiation to complete failure. As expected, the semi-crack fracture behavior was observed for the PC and its MWCNTs nanocomposites. It can be clearly seen that incorporation of MWCNTs has significant effects on the impact performance of the pure PC. The maximum load, deflection to maximum load and area of the force-deflection curve were greatly increased with increasing filler content.The value of the maximum load for the pure PC is about 209 N. Incorporation of MWCNTs causes a significant increase of ∼320% (884 N) at 1 wt% MWCNTs loading. This indicates that the PC nanocomposites are able to sustain much higher external force before being broken, and the behavior contributes to greater deflection. The increased filler content leads to higher impact force due to the particle interface react and form a tortuous fracture path The peak point on the maximum load curve corresponds to the radial fracture damage point. This point marks the onset of failure in the material as this initiation of damage induces a decrease in material stiffness, resulting in a drop in the load time profile. The drop in load marks the boundary line between two distinct phases, i.e. fracture initiation and fracture propagation. This implies that a higher force is needed to initiate a crack in the PC/MWCNT nanocomposite.For a semi-fracture system the total absorbed energy can be divided into two parts. The first is required for crack initiation, the elastically stored energy in the composite plate, which is released after maximum deflection by rebounding of the sample. The second is, the energy absorbed in the material available for crack propagation that consequently controls the extent of damage and residual strength. The amounts of total energy absorbed by falling weight impact, as measured by integration of the area underneath the force-deflection curves from . As expected, the PC nanocomposite exhibited a greater amount of energy absorption than the pure PC. Quantifiably, the nanocomposite with 1 wt% MWCNTs loading exhibited the largest amount of energy absorption, giving an improvement, than that of the pure PC. This improvement in impact failure energy is also an evidence of increase in energy dissipation capability of PC upon the addition of the MWCNTs. As a rule of thumb, the greater the energy dissipation capability of a system, the tougher it is! Also of note, with additional MWCNTs, the energies of initiation and the energy of propagation were also increased. In order to compare the reinforcement between two parts of the energies, the percentage of initiation energy for each sample was analyzed as shown in . As seen, the most part of the absorbed energy was consumed within the crack-initiation stage, whereas the crack propagation contributed a small proportion of energy consumption when the sample contained higher loading of MWCNTs. During impact the nanoparticles serve as stress concentrators in the matrix to build up a stress field around themselves. Because of the weak adhesion between the particles and the polymer matrix, debonding at the particle-matrix interface took place, leading to the release of the strain constraints at the crack tip and consequent massive plastic deformation, which consumed a large amount of energy shows representative digital images for the tested specimens of the pure PC and its nanocomposites. The patterns of damage in the entire configuration were presented. For the pure PC, the crack path was erratic and brittle fracture. The sample containing 0.1 wt% MWCNTs shows the similar model as in the pure PC. However, the samples containing 0.5 or 1.0 wt% MWCNTs exhibited plastic deformed yielding zone in the area where the falling weight striker hit, though the samples are fully penetrated. This again demonstrates visible evidence for dramatic improvement in the toughness of the PC. Dispersion of nanoparticles in a polymer matrix is essential for reinforcing impact performance of the material. When large aggregates are present the voids that are created by debonding are not stable and grow to a size where crack initiation occurs. The creation of stable free volume at the particle size level could lead to high energy adsorption by shear yielding and consequently high impact resistance revealed that uniform dispersion and distribution of MWCNTs in nanoscale was achieved in the PC matrix. It could be one reason for the reinforcement of impact properties of the PC.The PC and its nanocomposites were examined by means of an ‘in-house’ four-bar split Hopkinson Pressure Bar (SHPB) system for high strain rate performance. shows the stress-strain response curves for the PC and its MWCNTs nanocomposites at strain rates of about 3100 s−1, 3400 s−1, 3800 s−1 and 4000 s−1, respectively. It was seen that the stress-strain behavior for all materials including the pure PC exhibited an initial near-linear behavior where they yielded, followed by a transitional non-linear response where strain hardening takes place prior to a strain softening behavior. Addition of the nanofiller into the PC matrix resulted in significant changes in the flow and yield stress and the area under the curves for the resulting nanocomposites. These materials were also highly strain-rate-dependent.At high strain rates, the impact behavior of polymeric materials is very complex due to their temperature sensitivity, strain hardening and strain rate sensitivity depends on the applied strain rates. With increasing strain rate, the yield stress increased. Introduction of a very small amount of nanofiller material into the PC matrix resulted in a significant improvement in yield stress. In particularly, the nanocomposite with 1 wt% MWCNTs exhibited higher yield stress, giving an increase of ∼25% at 3100 s−1 strain rate. However, it was noted that the nano-reinforcement effect became weaker with increasing strain rate, especially for the PC containing higher MWCNTs contents. The yield stress for 1.0 wt% nanocomposite material was increased only by ∼13% at the highest strain rate of 4000 s−1. shows in terms of energy absorption for each material from beginning to the yield point. It can be seen that all nanocomposite materials exhibited a great amount of energy absorption than that of the pure PC. For example, for 1 wt% MWCNT nanocomposite the energy absorption increased by ∼15% at 3100 s−1 strain rate. Again, when the strain rate was increased from 3800 s−1 to 4000 s−1, 1 wt% MWCNTs nanocomposite did not show an obvious increment of the energy absorption. In order to understand this phenomenon, a comparison study for PC/clay nanocomposites was done at the same high strain rates. shows energy absorption for PC/clay nanocomposites at different strain rates. The energy absorption characteristic is similar in PC/MWCNT system. With increasing the strain rate or clay content the absorbed energy increased. However, the nano-reinforcement effect was also achieved even though for the PC containing higher percentage of 2 wt% clay at highest strain rate.For most polymeric materials it is common to find that high temperature levels can be rapidly induced when the material is being deformed at a higher strain rate, where the test is performed quickly After the SHPB tests, the microstructure of samples could be changed due to applied high speed impact. However, it is difficult to clearly observe the damaged specimens using SEM because of the limitation for preparing SEM specimens. In order to gain insight into the changes of microstructure in the PC and its nanocomposites, the density of the specimens before or after the SHPB tests was measured and the results are shown in . The results indicated that after the SHPB tests, the density of the pure PC and its nanocomposites decreased and the density drop with increasing strain rate in the nanocomposites was faster than that in the pure PC. This illustrates that more cracks formed in the PC nanocomposites during the SHPB tests. The decrease in density with increasing strain rate could result in the formation of a large amount of cracks during the SHPB tests. The more cracks form, the more energy dissipates. This could be the reason why the increase in impact energy with increasing strain rate in PC/MWCNT nanocomposite is greater than in the PC.PC and its MWCNTs nanocomposites with 0.1, 0.5 and 1 wt% content were assessed for their impact resistance behavior under dynamic loadings. The results from the IFIM tests indicate that MWCNTs were significantly effective in improving the impact resistance of the PC at very low filler loadings. In particular, the maximum load and the impact failure energy were dramatically increased by ∼320% and ∼350%, respectively, when only 1 wt% MWCNTs was incorporated. The performance of the PC and its nanocomposite at high strain rates of 3100 s−1, 3400 s−1, 3800 s−1 and 4000 s−1 examined by the SHPB showed that all the materials exhibited appreciable strain-rate sensitivity and the incorporation of MWCNTs lead to increase in yield stress and impact energy absorption. For 1 wt% MWCNT nanocomposite, the yield stress and impact energy increased greatly by ∼25% and ∼15%, respectively, at the high strain rate of 3100 s−1 as compared to the PC matrix. This improvement was understood by studying the changes in the densities of the materials before and after the SHPB tests. The decreased density illustrates that more cracks were formed in the PC nanocomposite during the SHPB tests, which could result in high energy dissipation. However, the nano-reinforcement effect at higher strain rate can have significant influences by involving the thermal-softening effect, especially for the PC nanocomposite containing higher MWCNT content. According to the comparison study, it would be better to use low thermal conductive reinforcement nanofiller for the fabrication of polymer nanocomposite materials for use in situations involving impact loadings such as crashworthiness of automobiles and armor development.Modeling the effect of temperature on the yield strength of precipitation strengthening Ni-base superalloysFor describing the temperature-dependent yield strength (TDYS) of precipitation strengthening Ni-base superalloys, a temperature-dependent model of Anti-Phase Boundary (APB) energy was developed in this study firstly. And then combining the proposed temperature-dependent APB energy model with the classical particle shearing theory, two temperature-dependent critical resolved shear stress models were developed for the weak and strong coupled dislocation pairs, respectively. Furthermore, the transition of dislocation motion mode from shearing to by-passing with increasing temperature was described theoretically by defining a temperature-dependent probability function of by-passed precipitate particles. In this way, a TDYS model for precipitation strengthening Ni-base superalloys was developed. The model contains the contributions of the precipitates, the grain boundary, the solid solution and the base metal at different temperatures, and their corresponding variations of the strengthening mechanisms with temperature. Moreover, the TDYS of seven typical precipitation strengthening Ni-base superalloys was predicted, and good agreement is obtained between the predicted results and the experimental data. The contribution of each mechanism to TDYS with increasing temperature was analyzed in this study. It indicates that the transition of dislocation motion mode from shearing to by-passing with increasing temperature will weaken the high-temperature yield strength of precipitation strengthening superalloys. Both grain refinement and solid solution strengthening are effective methods to reduce the probability of transition from particle shearing to by-passing, which can help maintain the high-temperature yield strength of superalloys. Using the proposed TDYS model, the optimal precipitate size to achieve maximum strength at different temperatures was analyzed. As temperature increases, the optimal precipitate size of superalloys decreases. And the optimal precipitate size at the most dangerous service temperature of the alloy can be determined by our model when designing a new superalloy.The high-temperature mechanical properties of various metallic materials have been widely investigated by many scholars (). As one of the most promising metallic materials, Ni-base superalloys have been applied in jet engines, nuclear power reactors and components in space technology (), due to their excellent high-temperature mechanical properties up to 800 °C (There are much efforts which have been paid to uncover the relationship between the mechanical properties and the microstructure of superalloys in recent years ( investigated the strength, fatigue life and fracture behavior of nickel-base superalloy PM 3030. proposed a microstructure-sensitive constitutive model for the inelastic behavior of superalloys. developed a numerical model, considering various microstructural factors, to predict the mechanical properties of Ni-base superalloys. have conducted a large amount of theoretical research and established a series of models to characterize the dislocation behavior in different superalloys. The above works are helpful to understand the mechanical behaviors of superalloys, but the theoretical modeling of yield strength at different temperatures for superalloys is still lacking. Previous work focused on characterizing the control mechanisms for the yield strength of superalloys at ambient temperature (). However, superalloys are always inevitably subject to high temperature environments during service, and temperature has significant influences on their mechanical behaviors. Many investigators have predicted the temperature dependence of yield strength of superalloys by substituting the experimental parameters at different temperatures into an ambient temperature model (). But some experimental parameters at different temperatures used in previous works are difficult to obtain, such as APB energy at different temperatures. More importantly, although the temperature-dependent material parameters have been considered in previous works, the corresponding variations of the strengthening mechanisms with temperature or the occurrence of some new mechanisms and their evolution with temperature have not been well quantified, which can significantly affect the TDYS of superalloys. examined the dislocation motion mode in IN100 at room temperature and 650 °C, respectively. pointed out that at room temperature the shearing strengthening model fits the experimental data satisfactorily up to the particle size of 400 nm in IN100. But the dominant deformation mode was found by to be Orowan by-passing mechanism verified by the TEM observations in IN100 at 650 °C for the mean precipitate sizes, 72.1 nm and 79.3 nm. reported that the transition of dislocation motion mode from shearing to by-passing occurs with the increase of temperature when particle size is larger than 63 nm in IN100. This transition was also observed in Ni-base superalloy M951 by . However, the quantitative effect of this transition on the TDYS has not been established. Therefore, to better understand the yield behavior of precipitation strengthening Ni-base superalloys, it is necessary to establish a theoretical model which considers the combined effects of each strengthening mechanism at different temperatures. The present study aims to quantitatively analyze the effect of temperature on the yield strength of precipitation strengthening Ni-base superalloys.APB forms on the slip plane when dislocations shear the ordered and coherent particles in superalloys. APB energy is especially important to predict the yield strength, while it is difficult to be experimentally measured (), especially at different temperatures. Many investigators suggested that APB energies can be calculated by theoretical models ( pointed out that the APB energy γAPB can be expressed by the following relationship:where S and a are the long-range order parameter and the lattice parameter, respectively. kB and Tc are the Boltzmann's constant and the critical ordering temperature, respectively. pointed out that the L12 alloys can be classified into two types: 1) type I with Tc<<Tm including Cu3Au, Ni3Fe, Ni3Mn and Ir3Cr; and 2) type II with Tc≥Tm including Ni3Si, Ni3Ga, Ni3Al, Ni3Ge and Zr3Al. Tm is the melting temperature of L12 alloys. The melting temperature of Ni3Al, the main precipitated phase of Ni-base superalloys, is 1688 K (, kB and Tc are temperature independent.The temperature-dependent a(T) is given as:where a(0) is the lattice parameter at absolute zero, α(T) is the temperature-dependent linear expansion coefficient.Many scholars have studied the relationship between the internal energy and ordered parameters of crystals (). The temperature-dependent long-range order parameter S(T) can be approximately expressed as (where Uc(0) and Uc(T) are the internal energy at absolute zero and temperature T, respectively. Uc(∞) is the internal energy at the stage of complete disorder., and taking T0 as an arbitrary reference temperature, then we can obtain:γAPB(T)=γAPB(T0)(1+∫0T0α(T)dT1+∫0Tα(T)dT)2(1−(Uc(T)−Uc(T0)Uc(∞)−Uc(T0)))The internal energy has the relationship with the isochoric specific heat as (where CV(T) is the isochoric specific heat of Ni3Al at temperature T. The temperature-dependent isochoric specific heat of Ni3Al can be obtained from handbooks or literature. To estimate the effects of temperature on the internal energy, the internal energy at different temperature can be expressed from Eq. Uc(T)=U0+∫0TCV(T)dTforT<TmUc(T)=U0+∫0TCV(T)dT+ΔHMforTm≤T<TBUc(T)=U0+∫0TCV(T)dT+ΔHM+ΔHBforT≥TBwhere U0 is the internal energy at absolute zero. TB is the boiling point of the material in Kelvin. ΔHM and ΔHB are the molar enthalpy of fusion at the melting point and the molar enthalpy of vaporization at the boiling point, respectively. In Eq. , it is an approximate method to estimate the temperature-dependent order parameter of crystal by internal energy (). For pure Ni3Al, it will lose its long-range order when approaching the critical ordering temperature Tc with a sudden change in internal energy. However, short-range order in precipitated phase γ′ still exists when temperature is higher than Tc until boiling ( pointed out that the vaporized material can be considered to be completely disordered. It means that pure Ni3Al can be considered to be completely disordered when temperature is higher than its boiling point. Therefore, it is assumed that the atoms are in complete disorder when the temperature is higher than the boiling point of crystal. However, the γ′ phase, as the main strengthening phase in the Ni-base superalloy, will dissolve into the disordered γ phase with increasing temperature below the melting point. And the structural disordering in the γ′ phase of superalloy will occur when temperature reaches the dissolving temperature of γ′ phase, and this was verified by also pointed out that this disordering has not been observed in the pure γ′ phase. The temperature dependence of γ′ volume fraction in superalloys has been examined experimentally (). The volume fraction of γ′ in superalloy can keep stable up to 800 °C and then significantly decreases as the temperature increases (). Thus, before reaching the dissolving temperature, the order parameter of the gamma prime phase in superalloy could be equivalent to that of the pure gamma prime phase. Substituting Eq. , and modified temperature-dependent order parameter by the normalized temperature-dependent volume fraction of γ′ phase in Ni-base superalloy, the temperature-dependent APB energy can be expressed as:γAPB(T)=γAPB(T0)f(T)f(T0)(1+∫0T0α(T)dT1+∫0Tα(T)dT)2(1−∫T0TCV(T)dT∫T0TBCV(T)dT+ΔHM+ΔHB)where f(T) and f(T0) are the volume fraction of gamma prime phase at temperature T and T0, respectively. In the model (Eq. ), α(T), CV(T), ΔHM and ΔHB can be obtained from material handbooks. The temperature-dependent APB energy model is physics-based and has no fitting parameters. The APB energy at different temperatures, which are not easy to be obtained by experiments, can be easily predicted by Eq. with reference to an easily-obtained APB energy. For convenience, the reference temperature is set as room temperature.In this study, to estimate the temperature dependent APB energy of the precipitated phase γ′, the internal energy in Eqs. should be that of the precipitated phase γ'. Tm, TB, α(T), CV(T), ΔHM and ΔHB in the model Eq. should be set as those of the precipitated phase γ'. The calculated values of the temperature dependent APB energy of the precipitated phase γ′ by Eq. will be used in the following sections.The yield strength of precipitation strengthening superalloys is mainly controlled by the shear mechanism of the precipitates, and therefore the strengthening increment caused by the γ’ precipitate shearing needs to be considered. The shearing hardening of precipitates is generally explained by the shearing of precipitates with APB and coherency effects (). The strengthening caused by the APB effect can be described by weak and strong pair coupling for different characteristics of dislocation motion. The CRSS increment due to weakly coupled dislocations can be expressed as (τAPB,weak=12(γAPBb)32(bdsfshearλ)12B−12(γAPBb)fshearwhere τAPB,weak, b, ds and fshear are the CRSS increment due to weakly coupled dislocations, Burgers vector, the diameter and volume fraction of sheared particles, respectively. B equals to 0.72 for spherical precipitates. λ is the line tension, λ=Gb22 for screw dislocations (), G is shear modulus. The transition from weak dislocation coupling to strong dislocation coupling will occur with increasing the size of precipitates ( have analyzed this phenomenon theoretically. Based on the theory by , the expression of CRSS increment for strong coupling is given by (τAPB,strong=12(Gbds)fshear120.72w(πdsγAPBwGb2−1)12w is a constant which reflects the repulsion between the pair of dislocations and approximately equals to one (In addition, when predicting the CRSS increment caused by APB at different temperatures, it is necessary to include the temperature dependence of APB energy, Burgers vector, shear modulus, particle size and volume fraction. Thus, we extend Eqs. to be temperature dependent, given as follows:τAPB,weak(T)=12(γAPB(T)b(T))32(4r0(T)fshear(T)G(T)b(T))12×0.72−12(γAPB(T)b(T))fshear(T)τAPB,strong(T)=12(G(T)b(T)2r0(T))fshear12(T)×0.72×(2πr0(T)γAPB(T)G(T)b(T)2−1)12where γAPB(T), G(T), r0(T), fshear(T) and b(T) are APB energy, shear modulus, average radius of precipitates, volume fraction of sheared particles and Burgers vector at temperature T, respectively., the temperature-dependent CRSS models are obtained for the weak and strong pair couplings as follows:τAPB,weak(T)=0.72γAPB32(T0)b2(T)(r0(T)fshear(T)G(T))12f(T)f(T0)(1+∫0T0α(T)dT1+∫0Tα(T)dT)3×(1−∫T0TCV(T)dT∫T0TBCV(T)dT+ΔHM+ΔHB)32−fshear(T)γAPB(T0)2b(T)×f(T)f(T0)(1+∫0T0α(T)dT1+∫0Tα(T)dT)2(1−∫T0TCV(T)dT∫T0TBCV(T)dT+ΔHM+ΔHB)τAPB,strong(T)=12G(T)b(T)2r0(T)fshear12(T)×0.72×[2πr0(T)γAPB(T0)G(T)b(T)2×f(T)f(T0)(1+∫0T0α(T)dT1+∫0Tα(T)dT)2(1−∫T0TCV(T)dT∫T0TBCV(T)dT+ΔHM+ΔHB)−1]12 established the mathematical expression of CRSS increment caused by coherency strains. The theoretical model proposed by is extended to be temperature-dependent as follow:τCoh(T)=AG(T)|ε(T)|32(r0(T)fshear(T)b(T))12for9πfshear(T)16<A|ε(T)|r0(T)b(T)<12where τCoh(T) and ε(T) are the temperature-dependent CRSS increment caused by coherency strengthening and linear misfit parameter, respectively. The linear misfit parameter can be expressed as ε(T)=2[aγ′(T)−aγ(T)]/[aγ′(T)+aγ(T)] (). A is a constant, A=3 for an edge dislocation and A=1 for a screw dislocation.Considering the effect of temperature on the dislocation motion resistance in precipitation strengthening Ni-base superalloys, the Kear-Wilsdorf (KW) locks are formed within the large γ′ precipitates under thermal activation ( suggested that thermally activated cross-slip pinning to form the KW lock is an additional strengthening mechanism active in primary and large secondary γ′ precipitates which with size larger than 300 nm. The yield strength increment caused by cross-slip-induced hardening can be expressed as (where Δσcs(T) is the yield strength increment by cross-slip-induced hardening, fcs is corresponding fraction of precipitates, σNi3Al(T) is the TDYS of pure Ni3Al, and Ci is the concentration of alloying element i in γ′ precipitates. dσ/dCi is the cross-slip-induced hardening constants for individual alloying elements which are assessed by For the simultaneous effect of APB, coherency and thermally activated cross-slip strengthening mechanisms to the total yield strength, the superposition of APB and coherency () with thermally activated cross-slip strengthening mechanisms can be expressed as:Δσshear(T)=M(τCohq(T)+τAPBq(T))1q+Δσcs(T)where M is the Taylor factor, M=2.6 for face center cubic polycrystalline materials ( for the CRSS increment due to weakly coupled dislocations and by Eq. for the CRSS increment due to strongly coupled dislocations. q is a parameter. suggested q=1.8 for the simultaneous effect of APB and coherency strengthening.As the particle size continually increases, the transition from precipitate shearing to precipitate by-passing will occurs (). The strengthening formalism for by-passed precipitates was developed by Orowan and extended by Ashby (). The by-passing strengthening σloop was quantified by the Ashby-Orowan equation (where floop is the volume content of by-passed particles, x¯=2r0 for spherical particles.Substituting the temperature-dependent parameters into Eq. , the temperature-dependent by-passing strengthening, σloop(T), is modified as:Δσloop(T)=0.3G(T)b(T)2r0(T)floop12(T)ln(r0(T)b(T))where floop(T) is the temperature-dependent volume fraction of by-passed particles.Both the change of particle size and temperature will cause the transition of strengthening mechanism (). The transition from shearing to by-passing with the increasing of temperature has been verified experimentally by many investigators (). It is generally believed that the occurrence probability of by-passing mechanism increases as the strength of the precipitated phase increases. It is widely investigated that single-phase strength of γ and γ’ are significantly influenced by temperature. In this work, to describe the mechanism transition of the dislocation motion mode under different temperatures, an explicit functional relation is proposed based on the temperature-dependent strength of precipitate and matrix.The temperature-dependent CRSS of γ′ phase and γ matrix are fully analyzed (). CRSS of γ’ increases with increasing temperature from 100 °C to 800 °C while that of γ decreases. The by-passing mechanism is more likely to occur when the strength of precipitation precipitates becomes much higher than that of matrix with increasing temperature. Based on the above analysis, the temperature-dependent probability function for the occurrence of Orowan by-passing is defined as:p(T)={σγ′(T)−σγ(T)2(σγ′(T)+σγ(T))whenσγ′(T)>σγ(T)0whenσγ′(T)<σγ(T)where p(T) is the temperature-dependent probability for the occurrence of Orowan by-passing. σγ′(T) and σγ(T) are the strength of γ′ phase and γ matrix at temperature T, respectively. Meanwhile, the probability for the occurrence of shearing is 1−p(T).The volume fraction of by-passed particles at temperature T is given as:where f(T) is the total volume fraction of particles at temperature T., the temperature-dependent volume fraction of by-passed particles can be obtained as:floop(T)={σγ′(T)−σγ(T)2(σγ′(T)+σγ(T))f(T)whenσγ′(T)>σγ(T)0whenσγ′(T)<σγ(T)And the temperature-dependent volume fraction of sheared particles is obtained:fshear(T)={σγ′(T)+3σγ(T)2(σγ′(T)+σγ(T))f(T)whenσγ′(T)>σγ(T)f(T)whenσγ′(T)<σγ(T), we can describe the transition from shearing to by-passing with increase of temperature theoretically.The strengthening mechanisms have been analyzed in deep for precipitation strengthening superalloys. The yield strength of precipitation strengthening superalloys (σys) is constituted by the yield strength of Ni matrix (σy0), grain boundary strengthening (ΔσD), solid solution strengthening (Δσss), second-phase precipitation shearing (Δσshear) and by-passing (Δσloop). There are several possible methods for the superposition of these strengthening increments, most of which assume the general form (where k is a fitting parameter between 1 and 2. predicted yield strength of Ni-base superalloys with different values of k, and pointed out that taking k to be significantly different from 1 resulted in considerable underestimation of the yield strength. The linear superposition of these strength terms from each strengthening mechanism is adopted by many scholars (). Then the TDYS of the precipitation strengthening superalloys is characterized by:σys(T)=σy0(T)+ΔσD(T)+Δσss(T)+Δσshear(T)+Δσloop(T)where ΔσD and Δσss are the strengthening increments due to grain boundary strengthening and solid solution strengthening, respectively. The temperature-dependent second-phase precipitation shearing (Δσshear) and by-passing (Δσloop) can be expressed by Eqs. , respectively. The temperature-dependent volume fraction of by-passed and particles can be estimated by Eqs. For the base metal, the yield strength is strongly temperature-dependent (). The TDYS of metal matrix is developed in our previous work (σy0(T)=[E(T)E(T0)(1−∫T0TCp(T)dT∫T0TmCp(T)dT)]12σy0(T0)where E(T) and Cp(T) are Young's modulus and isobaric specific heat at temperature T, respectively. E(T0) and σy0(T0) are Young's modulus and yield strength of metal matrix at reference temperature T0, respectively. Tm is the melting point of the metal matrix. The temperature-dependent model for the yield strength of metallic materials (Eq. ) is established based on a kind of equivalence between heat energy and distortional strain energy. It is assumed that the yielding of materials at temperature T occurs when the sum of the elastic deformation energy per unit volume and the corresponding heat energy reaches a certain value. In the model (Eq. ), the micro mechanisms whose influences on yield strength are irrelevant to temperature dependence have been considered in the model by the reference yield strength σy0(T0). The yield strength model can accurately predict the temperature-dependent yield strength until new micro mechanisms occur, which will evidently affect the yield strength. If new mechanisms occur, the corresponding mechanisms should be considered in the model. The model validation procedure have been carried out by our team and shows that the predicted results by the model are in good agreement with experimental results in the published papers. The model has been successfully applied to the high-temperature yield strength characterization of various metallic materials (, ΔσD represents the Hall-Petch relationship (): ΔσD=k/D, where k and D are the Hall-Petch constant and the grain size, respectively. For Ni-base superalloys, the Hall-Petch constant can be expressed as k=βGb (), where β is a temperature independent proportional factor. To combine the temperature effects, the temperature-dependent grain boundary strengthening can be described as:where D(T) is grain size at temperature T. k(T0), G(T0) and b(T0) are the Hall-Petch constant, shear modulus, Burgers vector and grain size at reference temperature T0.Solid solution strengthening was analyzed using Labusch's theory ( extended the Labusch's theory to multicomponent systems. suggested that the temperature dependence of solid solution strengthening for multicomponent alloys can be obtained by combining Labusch's solution model (where Δσss(T) is the solid solution strengthening term, ZL is a constant which equals to 1/550 (). α is a constant (3<α<16 for screw dislocations; α≥16 for edge dislocations). δi=a−1(da/dci), where a is the lattice parameter. ci represents the concentration of solute atoms i.ηi'(T)=ηi(T)/(1+|ηi(T)|/2), ηi(T)=G−1(T)(dG(T)/dci), ηi(T) is the modulus misfit and could also be given as ηi(T)=(Gi(T)−GB(T))/GB(T) (), where Gi(T) and GB(T) are the shear modulus of the solute and metal matrix at temperature T.) is established for precipitation strengthening superalloys in this study. The equations for the strengthening terms in Eq. are used to assess the strengthening effects caused by spherical precipitates in this study. It should be noted that the equations can also be used to assess the strengthening effects of cuboidal precipitates in some single crystalline Ni-based superalloys (). In this case, the parameter “mean diameter of particles” in the equations can approximately equal to the length of the diagonal of cuboidal precipitates. The model (Eq. ) considers the transition of dislocation motion mode from shearing to by-passing with the increase of temperature. It relates the TDYS to these basic parameters of the alloys, such as the temperature-dependent APB energy, the chemical composition, the TDYS of metal matrix, shear modulus, grain size, the size and volume fraction of precipitate. Most of the parameters could be conveniently obtained from the material handbooks or other researchers’ studies. Moreover, the TDYS of metal matrix and APB energy can be obtained with reference to that at an arbitrary temperature by our theoretical model.To verify the proposed temperature-dependent APB energy model (Eq. ), the temperature-dependent APB energies in Ni74.8Al21.9Hf3.3 are predicted, shown in . Besides, the predicted results of APB energies of Ni3Al with different element content of B determined from [00l] (110) tilt boundaries and [00l] twist boundaries with different misorientation angle θ at 1000 °C are shown in . In the theoretical calculations, the APB energy at 25 °C is set as the reference APB energy. The influence of element B on the APB energies of Ni3Al is considered in the model by the reference APB energy γAPB(T0). For solid materials, the value of CV(T) is approximately equal to that of Cp(T). The Cp(T) of most inorganic materials can be obtained from material handbooks (). The temperature-dependent isobaric specific heat of Ni3Al from 25 °C to the melting point (Tm = 1668 K) can be found in materials handbook (Cp(T)=88.492+32.217×10−3T(298.15K∼1668K)The heat capacity of Ni3Al from the melting point to boiling point is assumed to equal to the heat capacity at melting point, 142.23 J mol−1 K−1. The boiling point, melting heat and vapor heat of Ni3Al are approximately replaced by pure Ni due to the lack of related data (). For pure Ni, TB = 3187 K, ΔHM = 17472 J/mol and ΔHB = 369251 J/mol (). The temperature-dependent linear expansion coefficient of Ni3Al can be found in the study of , the APB energy at different temperatures can be predicted conveniently without any fitting process. Therefore, it is convenient to use the temperature-dependent APB energy model (Eq. ) to characterize the quantitative effect of temperature on the APB energy., the TDYS of seven precipitation strengthening Ni-base superalloys are predicted and shown in . The alloy composition of the seven superalloys is given in . In this study, the effect of these elements with the small content (lower than 0.1 at.% in superalloys) on the yield strength of the superalloys is ignored. The initial microstructures of the seven superalloys are shown in dp, ds and dt are the mean diameter of primary, secondary and tertiary γ′, respectively; fp, fs and ft are the corresponding volume fractions. In the case of IN100 and RR1000, γ′ shearing is the dominant feature for tertiary γ′ particles, and the by-passing of γ′ is the main mechanism for primary γ′ particles. The transition of dislocation motion mode from particle shearing to by-passing with increasing temperature occurs for secondary γ’. For RR1000, the microstructures at 14 and 119 mm to the rim of turbine disc were assumed to be coarse and fine grain zone (). The temperature-dependent Young's moduli of IN100, RR1000, GH4037, GH4033, GH4169, GH4080A and GH4738 are obtained from literature and material handbooks (. The shear moduli of these superalloys are calculated from the Young's moduli, with G=E2(1+ν) and ν=0.32(The TDYS of pure Ni is obtained by our previous work (Eq. ). For σy0(T), the temperature dependence of Cp(T) for pure Ni expressed as ({Cp(T)=19.083+23.497×10−3T(298.15K∼500K)Cp(T)=−251.166+356.439×10−3T+259.454×105T−2(500K∼631K)Cp(T)=467.194−678.737×10−3T(631K∼640K)Cp(T)=−385.698+404.225×10−3T+654.532×105T−2(640K∼700K)Cp(T)=−10.874+54.668×10−3T+56.476×105T−2−16.489×10−6T2(700K∼1400K)Cp(T)=36.192(1400K∼1726K)). Room temperature is chosen as the reference temperature, and σy0(T0)=103 MPa (For ΔσD(T), the Hall–Petch constant k(T0) for superalloys at room temperature is 750 MPaμm1/2 as reported by ), and its weak temperature dependence is ignored. For Δσss(T), the Young's modulus and Poisson's ratio of the solute and solvent are shown in ). The temperature dependence of Poisson's ratio is ignored (). The element atom fractions in the γ matrix and γ′ particle for IN100, in the γ matrix for RR1000 are given in , respectively. It is assumed that the chemical composition in the γ matrix for the other superalloys approximately equal to the composition of alloys, because of the low volume fraction of precipitate particles. The effect of solute elements on the cell parameters of Ni is obtained from a material handbook (For Δσshear(T), the APB energy at room temperature is 200 mJ/m2 for IN100 (), and 111 mJ/m2 are suggested for GH4037, GH4033, GH4169, GH4080A and GH4738 (). The temperature-dependence of APB energy can be calculated by Eq. . According to Vegard's law, the lattice parameters of γ and γ′ depend on the elemental compositions in each phase (aγ=3.524+0.110xCrγ+0.0234xCoγ+0.478xMoγ+0.444xWγ+0.441xReγ+0.179xAlγ+0.422xTiγ+0.700xTaγ+0.164xVγAoaγ′=3.570−0.004xCrγ′+0.208xMoγ′+0.194xWγ′+0.262xReγ′+0.258xTiγ′+0.500xTaγ′−0.060xVγ′Aowhere the term xij represents the mole fraction of component i in phase j. The aγ and aγ′ in IN100 are calculated from Eq. as aγ=0.3589 nm and aγ′=0.3597 nm. The linear misfit parameter between the matrix and the particle for IN100 is 0.22%. The linear misfit parameter in RR1000 is less than 0.07% (). The linear misfit parameters of GH4037, GH4033, GH4169, GH4080A and GH4738 are obtained by submitting the room-temperature yield strength into Eq. . The calculated linear misfit parameters are 0.515%, 0.757%, 1.361%, 0.558% and 0.531% for GH4037, GH4033, GH4169, GH4080A and GH4738, respectively.The transition of dislocation motion mode from shearing to by-passing in Ni-base superalloys RR1000 () are experimentally validated. The TDYS of RR1000 and IN100 were predicted and compared with the theoretical result by , respectively. Compared with the prediction assessed by , the variation tendency of yield strength with temperature predicted by our model is more consistent with the experimental values. For IN100 alloy, pointed out that the thermally activated cross-slip pinning is more likely activated when the precipitate size is larger than 300 nm. Considering that the cross-slip-induced hardening and transition of dislocation motion mode from particle shearing to by-passing with increasing temperature, the theoretical predicted results agree well with the experimental data of subsolvus IN100, as shown in . For the supersolvus IN100, the overestimated model predictions could be caused by the slight errors for the volume fraction and average diameter of tertiary γ’ () assume that the effects of temperature on yield strength is controlled by a fitting APB energy and shear modulus in this temperature range, but it does not consider the temperature variations of the other strengthening mechanisms. However, the strengthening mechanisms will change with the increase of temperature. In the theoretical predicted results without considering cross-slip hardening are much lower than the experimental data at high temperatures. Thus it is necessary to consider the variations of the strengthening mechanisms with temperature. As shown in , the predicted values of yield strength for GH4037, GH4033, GH4738, GH4169 and GH4080A by our model agree well with the experimental data. It indicates that the probability function defined in this study for the occurrence of Orowan by-passing can well describe the transition of dislocation motion mode from particle shearing to by-passing with increasing temperature. the influence of precipitation strengthening, solid solution strengthening and grain size on the TDYS of superalloys are analyzed. From , we can see that γ′ shearing of precipitates is the main contribution to the TDYS. However, the strength increment caused by γ′ shearing decreases quickly with the increase of temperature comparing with other contributors because the transition from γ′ shearing to γ′ by-passing will occur with increasing temperature. Maintaining the strength increment caused by γ′ shearing can keep the yield strength of precipitation strengthening superalloys with increasing temperature. As shown in , the amount of the Δσss(T) and ΔσD(T) in RR1000 with fine grain is much higher than that in GH4033. And compared with GH4033, the contribution of γ′ shearing strengthening still maintains at a high level with increasing temperature in RR1000, shown in . Both the solid solution and grain boundary strengthening are effective ways to reduce the probability of transition from particle shearing to by-passing, which can help maintain the high-temperature yield strength of superalloys. In addition, the volume fraction and precipitate size of γ′ have great effects on the yield strength (). Increasing the volume fraction of γ′ can significantly increase the yield strength ( examined the strongest particle sizes in RR1000 with unimodal γ′ precipitates at 600 °C and 700 °C. pointed out that, considering all of the precipitates to be sheared, the optimal precipitate size of RR1000 increases with the increase of temperature and volume fraction, as shown in , the elastic moduli at different temperatures are replaced by that at ambient temperature. The above simplification makes the calculated results of optimal precipitate size do not agree with the fact. Considering the influence of elastic moduli at different temperatures, the contrary tendency of optimal precipitate size with temperature was found comparing with Collins and Stone's work (. It indicates that it is important and necessary to consider the effect of temperature on the mechanical behaviors of superalloys. There are three major factors that influence the optimal particle size: temperature-dependent elastic modulus, temperature-dependent APB energy and volume fraction of γ’. Using the temperature-dependent parameters, the optimal particle size of γ′ to achieve maximum strength for 20 vol% γ′ in RR1000 at different temperatures is obtained when the transition from weak to strong dislocation coupling occurs, as shown in . The same method was also used to calculate the optimal particle size of γ′ in GH4037. The temperature-dependent optimal precipitate sizes for RR1000 and GH4037 precipitation strengthening superalloys are shown in . The higher the temperature is, the faster the optimal precipitate size decreases. To maintain the yield strength at high temperatures, the optimal precipitate size to achieve maximum strength should be decreased. On the other hand, the optimal precipitate sizes at different APB energies are shown in taking GH4037 as an example. The optimal precipitate size decreases quickly with the increase of APB energy.A theoretical model is developed to describe the TDYS of precipitation strengthening superalloys, and is applied to study the variations of the strengthening mechanisms with temperature in these superalloys. The following conclusions are summarized:A new temperature-dependent APB energy model, free of fitting parameters, is developed for calculating the formation energy of APB. The APB energy model provides a simple and practical method to obtain the temperature-dependent APB energy.The present model considers variations of the strengthening mechanisms with temperature, and can well predict the TDYS of precipitation strengthening superalloys.The contribution of each strengthening mechanism with increasing temperature to the TDYS of precipitation strengthening superalloys is first analyzed using the proposed model. The transition of dislocation motion mode from shearing to by-passing with the increase of temperature will weaken the high-temperature yield strength of these alloys. Meanwhile, both grain refinement and solid solution strengthening are effective methods to reduce the probability of transition from particle shearing to by-passing.The contrary tendency of optimal precipitate size with the increase of temperature was found comparing with the reported work. The optimal precipitate size to achieve maximum strength decreases with increasing temperature. The optimal precipitate size at the most dangerous service temperature of the alloy can be determined by our model when designing a new superalloy. Trans. Nonferrous Met. Soc. China 22(2012) 2379 2388 High temperature tensile properties of laser butt-welded plate of Inconel 718 superalloy with ultra-fine grains QU Feng-sheng 1 , LIU Xu-guang 1 , XING Fei 1 , ZHANG Kai-feng 2 1. School of Materials Science and Engineering, Chongqing University, Chongqing 400044, China; 2. College of Materials Science and Engineering, Harbin Institute of Technology, Harbin 150001,China Received 9 July 2012; accepted 8 August 2012 Abstract: For successfully forming multi-sheet cylinder sandwich structure of Inconel 718 superalloy, high temperature tensile properties of laser butt-welded plate of Inconel 718 superalloy were studied. The experiment results show that tensile direction has great effect on elongation of the laser butt-welded plate. Under conditions of transverse direction tension, the maximum elongation reaches 458.56% at 950 °C with strain rate of 3.1× 10 4 s 1 , in which the strain rate sensitivity value m is 0.352 and the welding seam is not deformed. Under conditions of longitudinal direction tension, the maximum elongation is 178.96% at 965 °C with strain rate of 6.2× 10 4 s 1 , in which m-value is 0.261, and the welding seam contributes to the deformation with the matrix. The microstructure in as-welded fusion zone is constituted of austenite dendrites and Laves phase precipitated in interdendrites. After longitudinal direction tension, a mixed microstructure with dendrite and equiaxed crystal appears in the welding seam due to dynamic recrystallization. After high temperature deforming, many -phase grains are transformed from Laves phase grains but a small part of residual Laves phase grains still exist in the welding seam. The deformation result of multi-sheet cylinder sandwich structure verifies that high temperature plasticity of the laser butt-welded plate can meet the requirement of superplastic forming. Key words: Inconel 718 alloy; laser welding; high temperature plasticity; microstructure 1 Introduction Inconel 718, an aging hardening Ni Cr Fe superalloy, has perfect mechanical properties, excellent anti-fatigue performance and corrosion resistance below 650 °C [1,2]. Therefore, it is widely used for aviation, aerospace, petroleum, chemicals, energy and other fields [3]. Multi-sheet cylinder sandwich structures of Inconel 718 superalloy are important heat resisting and shielding structures in high-speed vehicle [4]. However, the severe tendency of strain hardening of Inconel 718 alloy limits the sheet size to meet the size for multi-sheet cylinder sandwich structures [5]. The bending plate and butt-welding technology can make the small-size sheet splice the heavy-size sheet with homogeneous thickness of wall and shorter manufacturing cycle. Thus the high temperature plasticity of Inconel 718 superalloy with butt-welding seam has great influence on the manufacture of multi-sheet cylinder sandwich structure. In recent years, research works on laser butt-welded plate of Inconel 718 superalloy have been focused on the effect of heat treatment process on the microstructures and mechanical properties of the welded seam [1,6,7]. JANAKI RAMA et al [6] studied the microstructure and tensile properties of Inconel 718 pulsed Nd-YAG laser welds by heat treatment technology. GAO et al [7] studied the microstructures and high temperature mechanical properties of electron beam welded Inconel 718 superalloy thick plate, and the temperature of mechanical test was 650 °C. And the research works on superplasticity of a laser-welded Ti-alloy joint were reported [8 10]. There are few reports about high temperature plasticity and microstructures development of laser butt-welded plate of Inconel 718 superalloy beyond 900 °C. In this work, the high temperature tensile properties and microstructures development from 950 °C to 980 °C for laser butt-welded plate of Inconel 718 superalloy were studied. At last, the experiment results were applied to fabrication of the multi-sheet cylinder sandwich structure of Inconel 718 superalloy. Foundation item: Project (20102302120002) supported by the Research Fund for the Doctoral Program of Higher Education of China Corresponding author: QU Feng-sheng; Tel: +86-23-18696586956; E-mail: [email protected] DOI: 10.1016/S1003-6326(11)61474-X QU Feng-sheng, et al/Trans. Nonferrous Met. Soc. China 22(2012) 2379 2388 2380 2 Experimental Test material was Inconel 718 superalloy sheet with ultra-fine grains. The process for the ultra-fine grains: (1050 °C, 0.5 h) with successive water quenching, 50% cold rolling + (890 °C, 10 h) -phase precipitation treatment+30% cold rolling+(950 °C, 3 h) recrystallization annealing. The grain size is ASTM13 14 as shown in Fig. 1. The chemical composition is listed in Table 1. Table 1 Chemical composition of Inconel 718 alloy (mass fraction, %) C Si Mn Ni 0.067 0.08 0.03 53.32 Cr Fe Co Ti 19.26 16.42 0.05 1.13 Al Mo Nb S 0.55 3.26 5.73 0.005 P Mg Cu B 0.008 <0.01 0.072 0.006 Fig. 1 Microstructure of Inconel 718 superalloy with ultra-fine grains The laser butt-welding of Inconel 718 superalloy was carried out in a cross-flow CO 2 continuous laser generator (DC030 type), by the means of one-side welding with two-side formation technology. The welding parameters of Inconel 718 superalloy are listed in Table 2. High temperature deformation was carried out in an Instron5500R universal testing machine with an electrical resistance furnace with error less than ±0.1 °C. The high temperature tensile tests were conducted in two tensile directions: transverse direction and longitudinal direction, as shown in Fig. 2. The m-values were measured in a Gleeble 1500 thermal simulation testing machine by Backfen’s method. The microstructures of welded joint and base metal before and after tension were investigated by an OLYMPNS-GX51 optical microscope and an FEI QAUANTA 2000 scanning electron microscope. Element contents of precipitated phases and dendrites in the weld fusion were estimated by image analysis software. Table 2 Laser butt-welding parameters of Inconel 718 superalloy with ultra-fine grain Power/W Welding speed/ (mm·min 1 ) Defocusing amount/mm Shielding gas/(L·min 1 ) 900 1400 1 0.6 Fig. 2 Schematic diagram of Inconel 718 butt-welded plate tensile test 3 Results and discussion 3.1 High temperature tensile properties of laser butt-welded plate Specimens after high temperature tension at 950 980 °C with strain rate of 3.1× 10 4 s 1 are shown in Figs. 3(a) (transverse direction) and (b) (longitudinal direction). Curves of elongation—stress in transverse and longitudinal directions are shown in Figs. 4(a) and (b), respectively. The values of elongation, peak flow stress and strain rate sensitivity(m-value) and thickness of the weld seam are listed in Table 3. It can be seen from Fig. 3(a) that the specimens exhibit bamboo-shape after transverse direction tension and the m-values are all higher than 0.3. When the m-value is higher than 0.3 or the elongation is over 200%, the material exhibits superplasticity [11]. It is conducted that the deformation in this experiment is located at base material rather than welding seam, and this indicates that the strength of welding seam is higher than that of base metal in the test. From Fig. 1, it can be seen that the grains in base material are with grain size of ASTM13 14, which has superplasticity under the test condition. The major deformation mechanism is grain boundary sliding, with the coordination of dislocation slip [5]. But in the laser welding seam, it is as-cast microstructure in non-equilibrium state and Laves phase QU Feng-sheng, et al/Trans. Nonferrous Met. Soc. China 22(2012) 2379 2388 2381 distributes in interdendritic region of fusion interior and boundary, and the details will be discussed in Section 3.2. The as-cast microstructure at the welding seam cannot meet the superplasticity requirement of equiaxed grain size lower than 10 m for Inconel 718 alloy. It cannot exhibit the extremely low flow stress as that with superplasticity. In addition, Laves phase occurring in microstructures of the welding seam increases the flow stress in the welding seam. This is because Laves phase is a topologically close-packed phase with high-temperature strength and high creep resistance, which makes the flow stress in welding seam higher than that of the base metal and increases the creep resistance of the welding seam. In Fig. 3(b), it shows that under the conditions of longitudinal direction tension, welding seam and base material contribute to deformation. And the specimens do not exhibit the bamboo-shape but neck down. From Fig. 3(b) and Fig. 4(b), it shows that the elongation Fig. 3 Specimens in tensile test at different temperatures with strain rate of 3.1× 10 4 s 1 : (a) Transverse direction tension; (b) Longitudinal direction tension Fig. 4 Curves of elongation—flow stress of specimens with strain rate of 3.1×10 4 s 1 at different temperatures: (a) Transverse direction tension; (b) Longitudinal direction tension Table 3 Values of elongation, peak flow stress and m of specimens with strain rate of 3.1×10 4 s 1 at different temperatures Tensile mode Deforming temperature/°C Elongation/% Peak flow stress/MPa m-value Thickness of weld seam/mm 950 458.56 56.47 0.352 3.05 965 450.12 51.82 0.346 3.02 Transverse direction 980 285.92 49.60 0.318 3.06 950 148.96 85.70 0.237 2.03 965 174.78 78.56 0.252 1.86Longitudinal direction 980 158.95 71.70 0.247 1.92 QU Feng-sheng, et al/Trans. Nonferrous Met. Soc. China 22(2012) 2379 2388 2382 dramatically reduces but the flow stress significantly increases under the conditions of longitudinal direction tension. This is because the geometric position of the welding seam leads to the welding seam deforming. As shown above, the welding seam cannot exhibit superplasticity, which leads to the elongation decreasing and the flow stress increasing. And these states can be validated by the flow stress, m-value and thickness of the welding seam. Photos of tensile specimens at 965 °C with different strain rates in transverse and longitudinal direction tensions are shown in Figs. 5(a) and (b), respectively. And Table 4 shows values of the elongation, peak flow stress, m and thickness of the welding seam. From Table 4, it can be seen that within the range of strain rate of 3.1× 10 4 1.8× 10 3 s 1 in transverse direction tension, the values of elongation are over 260%, and m-values are higher than 3.0. With the strain rate increasing, the flow stress increases but the elongation decreases. In addition, the flow stress in the welding seam is higher than that of Fig. 5 Tensile specimens at 965 °C with different strain rates in transverse direction tension (a) and longitudinal direction tension (b) Fig. 6 Curves of elongation—flow stress of specimens at 965 °C with different strain values in transverse direction tension (a) and longitudinal direction tension (b) Table 4 Values of elongation, peak flow stress and m of Inconel 718 superalloy in tension at 965 °C with different strain rates Tensile mode Strain rate /s 1 Elongation/% Peak flowing stress/MPa m-value Thickness of weld seam/mm 3.1×10 4 450.12 51.82 0.346 3.01 6.2×10 4 316.70 62.40 0.327 3.02 Transverse direction 1.8×10 3 263.26 74.50 0.308 3.04 3.1×10 4 174.78 78.56 0.252 1.86 6.2×10 4 178.96 92.40 0.261 1.90 Longitudinal direction 1.8×10 3 142.82 112.60 0.223 1.98 QU Feng-sheng, et al/Trans. Nonferrous Met. Soc. China 22(2012) 2379 2388 2383 base material. For longitudinal direction tension, the largest elongation reaches 178.96% at the strain rate of 6.2× 10 4 s 1 . The flow stress increases with the strain rate increasing. The test results show that at 965 °C and in the range of strain rate 3.1× 10 4 6.2× 10 4 s 1 , the butt-welded plate exhibits excellent superplasticity, which can meet the forming requirements of the part. 3.2 Effect of high temperature tension on microstructures evolution 3.2.1 Microstructures of as-welded seam Figure 7 shows the microstructures of fusion zone in as-welded specimens. It is found that the dendrites in fusion interior are fine and equiaxed with grain size of ASTM15 16 (Fig. 7(a)), and at region adjacent to the fusion boundary, the dendrites are slightly coarser and columnar with grain size of ASTM8 9 ((Fig. 7(b)). According to the theory of morphological stability originally developed by MULLINS and SEKERKA [12], the temperature gradient (G) and the growth rate (R) are very important factors for determining the solidification morphology and grain size. The thermal gradients in a weld pool are steeper at regions close to fusion boundary than those in weld interior. The steeper thermal gradient prevailing at the fusion boundary is responsible for the columnar dendrite growth in a direction opposite to the heat extraction direction. Towards the weld center, Fig. 7 Microstructures of fusion zone (as-welded condition): (a) Very fine equiaxed dendrites in weld interior; (b) Columnar dendrites adjacent to fusion boundary however, the thermal gradient is not as steep, which in combination with the very rapid cooling rates of laser beam welding, results in significant undercooling (low G/R), leading to the formation of fine equiaxed dendrites. The gain size in heat affected zone (HAZ) grows gently to ASTM10 11 due to heat input in laser welding, as shown by the arrow in Fig. 7(b). The SEM microstructures of the weld fusion zone are shown in Figs. 8(a) (at weld center) and (b) (adjacent to fusion boundary). A number of precipitated phases occur in the weld fusion zone. The precipitated phases in the weld center distribute discretely. But the distribution of precipitated phases adjacent to the fusion boundary is slightly regular and directional. The element contents of dendrite and precipitated phase in weld joint measured by the EDAX spectrum analysis are listed in Table 5. From Table 5, it can be seen that the precipitated phase is rich in Nb, Mo and Ti, and poor in Ni, Cr and Fe, compared with the base material. The dendrite core is Fig. 8 SEM microstructures of fusion zone (as-welded condition): (a) In weld interior showing fine discrete Laves particles; (b) Showing a certain ordination Laves particles adjacent to fusion boundary Table 5 Element content of dendrite and precipitated phase in weld joint Mass fraction/% Phase Ni Fe Cr Nb Mo Ti Al Dendrite 54.63 19.16 19.51 2.82 2.72 0.65 0.51 Precipitate 48.55 14.39 16.40 14.58 3.82 1.49 0.78 QU Feng-sheng, et al/Trans. Nonferrous Met. Soc. China 22(2012) 2379 2388 2384 poor in Nb, Ti and Al. The element fraction characteristic shows that the precipitated phase is Laves phase which presents in the cast and welded products of Inconel 718 superalloy. Inconel 718 superalloy is precipitation strengthened primarily by (Ni 3 Nb). The sluggish ageing kinetics of precipitation is also found to be beneficial to material’s weldability [13]. However, since niobium is a high concentration refractory element, it tends to segregate during the solidification process, as a result of the fact that some desirable phase, like Laves phase [13], can form in the welding fusion. Laves phase is a hexagonally close packed phase in the form (Ni, Fe, Cr) 2 (Nb, Mo, Ti). And the forming process can be expressed by the following formulations (1) and (2) [14]. Liquid Liquid enriched with Nb+ (1) Liquid enriched with Nb +Laves (2) Laves phase is a brittle intermetallics and is topologically close-packed phase. For Inconel 718 supperalloy, the atoms with larger radius are Nb, Ti and Mo, and the minor ones are Fe, Ni and Cr. The crystal structures are complicate, and the coordination numbers reach 14 16 [15]. Laves phase provides the conditions for nucleation and crack growth, which severely impairs the plasticity of Inconel 718 at room temperature. A small amount of granular or small block Laves phase existing in Inconel 718 has little influence on mechanical properties; however, when the content of Laves phase is over 2% 3%, it harms the strength and plasticity of the alloy at room temperature, and impairs 20% strength and 60% plasticity [15]. At 650 °C and 620 MPa, the rupture life and rupture elongation are reduced by 60%, which exhibits the weakening effect of Laves phase. Furthermore, Laves phase occupies plenty of Mo, Ti and Ni, which can also affect the solid solution strengthening and decrease the number of and to weak the effect of precipitation strengthening. From the analysis of Laves phase in the welding seam, it can be found that there are (5.0±0.4)% Laves phase in the welding seam. In addition, Laves phase has high strength and high creep resistance at high temperature [16]. For instance, the yield strength of NbCr 2 at 1200 °C reaches 600 MPa. And this is the key reason why the welding seam is free from deformation in transverse direction tension and the flow stress in longitudinal direction tension remarkably increases. 3.2.2 Microstructures in welding seam after high temperature tension Metallographic structures of weld interior and boundary in transverse direction tension at 965 °C, strain rate of 3.1× 10 4 s 1 are shown in Figs. 9(a) and (b). From Figs. 9(a) and (b), the grains in welding seam are still dendritic, but grow a little. The grain size in weld interior is ASTM13 14 and a small part is ASTM11, and the columnar crystals in fusion boundary are coarser than grain size of ASTM6 because high temperature improves atomic activity. In addition, the grains in heat affected zone grow up with grain size of ASTM10. Microstructures of weld interior and boundary in longitudinal direction tension at 965 °C and strain rate of 3.1× 1 4 s 1 are shown in Figs. 9(c) and (d). Compared with Figs. 9(a) and (b), the microstructures in longitudinal direction tension change more obviously and the equiaxed grains with grain size of ASTM10 occur in welding seam, as shown by arrow. This is because the microstructures of the welding seam cannot perform superplasticity under the condition of longitudinal direction tension. During the high temperature deformation, on one hand, the dislocation density increases, on the other hand, the forming of subgrains resulting from dislocation dipole cancels, and the polygonization of dislocation cell wall and subgrains merging are conducted by thermal activation. For nickel alloy, a low stacking fault energy metal, it has large extended dislocation width which leads to difficulty to gather characteristic dislocation. Therefore, there is rarely cross-slip dislocation and climbing dislocation. The dynamic recrystallization occurs till the dislocation accumulating to a certainty degree. Thus, the equiaxial grains will occur at fusion zone after longitudinal direction tension. The microstructures of specimens in fracture are shown in Fig. 9(e). The grains in fracture at fusion zone are almost equiaxial and the cavities occur. This is because in high temperature tension, the dynamic recrystallization performs continuously resulting in a large number of equiaxial grains. The transformation from Laves phase to phase at grain boundary is shown in Fig. 10, and the residual Laves phase grains prevent the growth of equiaxial grains. With a larger deformation, especially the strain rate increasing at fracture leads to the crystallization nuclei for dynamic recrystallization increasing, and generates a large number of fine grains. The fine exquiaxed grains provide the possibility of superplasticity. And at the strain rate of 3.1× 10 4 s 1 , the fine equiaxed grains lead to boundary sliding, which is the main deformation mechanism of superplasticity in Inconel 718 superalloy. In addition, the larger flow stress in longitudinal direction tension leads to the strain rate increasing near the fracture. This results in the cavities increasing, and when the cavities are continuous, the specimen fractures. Moreover, the place where Laves phase grains occur and phase grains precipitate at the welding seam supplies nucleation and growth of the cavities. Figures 10(a) and (b) show the microstructures in weld interior and adjacent to boundary in transverse direction tension at 965 °C and strain rate of 3.1× 10 4 s 1 , respectively. From Figs. 10(a) and (b), it can be seen that QU Feng-sheng, et al/Trans. Nonferrous Met. Soc. China 22(2012) 2379 2388 2385 respectively. From Figs. 10(a) and (b), it can be seen that the morphology of precipitated phases changes a lot and there are a number of needle-like precipitated phases in the welding seam, with a spot of residual globosity phase. The microstructures of the weld fusion zone in longitudinal direction tension at 965 °C and strain rate of 3.1× 10 4 s 1 are shown in Figs.10(c) (at weld center) and (d) (at region adjacent to fusion boundary). In addition, the equiaxed grains in longitudinal direction tension can be obviously found. And the elements in dendrite, needle-like precipitated phase and globosity precipitated phase can be obtained by EDAX, as shown in Figs. 10(a), (b) and (c), respectively. Table 6 shows the element contents of the dendrite, needle-like and globosity precipitated phases by EDAX analysis. From the results of EDAX spectrum, the Nb content of the dendrite in the weld fusion zone increases to 4.83%. The Nb content is 5.73% in base material, and the Nb content of needle-like precipitated phase in the weld fusion is 8.03%. This indicates that the precipitated phase is phase. The phase (Ni 3 Nb) with orthogonal and ordered structure is stable phase of the metastable phase and precipitates in the temperature range of 860 995 °C [5]. The precipitation of phase at grain boundary can prevent the growth of grains in Inconel 718 alloy and promote fine and uniform grains, which increase the strengthening and toughening of the alloy. In addition, an appropriate amount of phase is in favor of eliminating sensitivity of the nick. In the weld metal solidification, the interdendritic regions become rich in Nb with the Nb content in the range of 6% 10% or more, and the Nb content in regions where Laves phase grains form is more than 10%. Thus, at the end of solidification, there will be the regions around Laves phase grains in the weld metal with sufficient Nb content where phase grains can precipitate. However, the rapid cooling rates through the phase precipitation range during welding metal preclude the phase precipitation from these regions. As shown in Fig. 10, a majority of Laves phase grains transform to needle-like phase grains because of enough forming time for phase precipitation in high temperature tension. Firstly, phase grains precipitate Fig. 9 Microstructures of welding seam after tension at 965 °C and strain rate of 3.1× 10 4 s 1 : (a) Weld interior in transverse direction tension; (b) Fusion boundary in transverse direction tension; (c) Weld interior in longitudinal direction tension; (d) Fusion boundary in longitudinal direction tension; (e) Fracture in longitudinal direction tension QU Feng-sheng, et al/Trans. Nonferrous Met. Soc. China 22(2012) 2379 2388 2386 around Laves phase grains which are rich in Nb element. And then the residual Nb element diffuses to the dendrites, which can also relieve segregation of Nb. By analyzing element contents in Table 6, it shows that the globosity phase is still Laves phase. This is attributed to the fact that whether Laves phase can be dissolved easily or difficultly depends on its Nb content and particle size. Coarser Laves phase grains with higher Nb content require higher temperature and longer soaking time for dissolution. The tensile direction has no influence on the contents of phases and elements. With the analysis by image software, the content of Laves phase in welding seam is 0.8%±0.07% after tension, which is within the safety range. The content of phase in transverse direction tension is 7.0% 8.3%, and in longitudinal direction tension is 6.4% 6.9%. The content of phase is decreased by deformation. 3.2.3 Microstructures of base material after tension The microstructures of base material after high temperature tension at strain rate of 3.1× 10 4 s 1 are shown in Figs. 11(a) (at 965 °C in transverse direction tension) and (b) (at 965 °C in parallel welding seam tension), respectively. Compared with microstructures of base material before high temperature tension, the grains of base material after high temperature tension grow gently, and the grain size is ASTM12 13, and the cavities appear in the base material. In general, the cavities occur at grain boundary and phase boundary during the superplastic deformation. And the temperature for tensile test is within the temperature range of precipitation of phase. On one hand, phase at grain boundary can prevent the growth of the grain; on the other hand, when the grain boundary slides, the particles at the boundary will not coordinate and lead to a large number of cavities. The microstructures of base material at 965 °C and strain rate of 3.1× 10 4 s 1 are shown in Figs. 12(a) (in transverse direction tension) and (b) (in longitudinal direction tension), respectively. In two situations, a large number of cavities are formed. The experiment results were applied to fabrication of multi-sheet cylinder sandwich structure of Inconel 718 superalloy. The photos of the application are shown in Fig. 13. Fig.10 SEM images of base material after tension at 965 °C and 3.1× 10 4 s 1 : (a) Weld interior in transverse direction tension; (b) Weld boundary in transverse direction tension; (c) Weld interior in longitudinal direction tension; (d) Weld boundary in longitudinal direction tension Table 6 Elements content of dendrite, precipitated phase after superplastic tension Mass fraction/ % Phase Ni Fe Cr Nb Mo Ti Al Dendrite 53.04 18.24 18.65 4.83 3.42 1.11 0.6 Needle-like precipitated phase 52.23 16.94 17.01 8.03 2.84 1.48 0.49 Spherical precipitated phase 47.51 15.86 15.63 14.73 3.94 1.67 0.66 QU Feng-sheng, et al/Trans. Nonferrous Met. Soc. China 22(2012) 2379 2388 2387 Fig. 11 Microstructures of base material after tension at strain rate of 3.1× 10 4 s 1 : (a) At 965 °C in transverse direction tension; (b) At 965 °C in longitudinal direction tension Fig. 12 SEM images of microstructures of base material at strain rate of 3.1× 10 4 s 1 at 965 °C: (a) In transverse direction tension; (b) In longitudinal direction tension Fig. 13 Application in multi-sheet cylinder sandwich structure of Inconel 718 superalloy: (a) Welding seam; (b) Top view of product 4 Conclusions 1) In transverse direction tension performed in temperature range of 950 °C 980 °C and strain rate of 3.1× 10 4 s 1 , the largest elongation of laser butt-welded plate is 458.56% at 950 °C, with a unique deformation region in base material. In longitudinal direction tension, the elongation reduces dramatically but the flow stress increases significantly. The largest elongation of 178.96% occurs at 965 °C and strain rate 3.1× 10 4 s 1 . The flow stress of the welding seam is higher than that of base material in the experiments. 2) The dendrites in as-welded welding interior are fine and equiaxed but in region adjacent to fusion boundary are coarser and columnar. And there are Laves phase grains formed in welding process. 3) The microstructures of the welding seam in transverse direction tension at 965 °C and strain rate 3.1× 10 4 s 1 preserve the characteristics in as-welded material, with a little growth. Under the same condition in longitudinal direction tension, the equiaxed grains form due to the dynamic recrystallization. In addition, a majority of Laves phase grains in the welding seam is transformed to phase grains after high temperature tension. Therefore, deformation is favorable to improving QU Feng-sheng, et al/Trans. Nonferrous Met. Soc. China 22(2012) 2379 2388 2388 the microstructures of the welding seam. 4) The cavities are formed in deformation of base material because the superplastic deformation mechanism of base material is grain boundary sliding, which leads to the formation of the cavities. References [1] SGOBBI S, ZHANG L, NORRIS J, RICHTER K H , LOREAU J H. High powder CO 2 and Nd YAG laser welding of wrought Inconel 718 [J]. Journal of Materials Process Technology, 1996, 56: 333 345. [2] WANG Yan, SHAO Wen-zhu, ZHEN Liang. Dissolution behavior of phase and its effects on deformation mechanism of GH4169 alloy [J]. The Chinese Journal of Nonferrous Metals, 2011, 21(2): 341 342.(in Chinese) [3] KONG Yong-hua, HU Hua-bin, LI Long, CHEN Guo-sheng, ZHU Shi-gen. Study on the microstructures and properties of GH4169 alloy by different forging technology [J]. Rare Metal Materials and Engineering, 2011, 40(S2): 225 226. (in Chinese) [4] QU Feng-sheng, LU Zhen, XING Fei, ZHANG Kai-feng. Study on LBW/SPF technology of multi-sheet cylinder sandwich structure for inconel 718 Superalloy [J]. Journal of Sichuan University: Engineering Science Edition, 2102,44(3): 185 186. (in Chinese) [5] LU Hong-jun. Study of untra-fine grain processing and superplastic forming of GH4169 superalloy sheet [D]. Harbin: Harbin Institute of Technology, 2003. (in Chinese) [6] JANAKI RAMA G.D, ENUGOPAL REDDYA A V, PRASAD RAOB K, REDDYC G M, SARIN SUNDAR J K. Microstructure and tensile properties of Inconel 718 pulsed Nd-YAG laser welds [J]. Journal of Material Processing Technology, 2005, 167: 73 82. [7] GAO Peng, ZHANG Kai-feng, ZHANG Bing-gang, JIANG Shao-song, ZHANG Bao-wei. Microstructures and high temperature mechanical properties of electron beam welded Inconel 718 superalloy thick plate [J]. Transactions of Nonferrous Metals Society of China, 2011, 21: s315 s322. [8] CHEN S H, HUANG J H, CHENG D H, ZHANG H, ZHAO X K. Superplastic deformation mechanism and mechanical behavior of a laser-welded Ti 6Al 4V alloy joint [J]. Materials Science and Engineering A, 2012, 541: 110 119. [9] CHENG Dong-hai, HUANG Ji-hua, CHEN Yi-ping, HU De-an. Microstructure evolution characterization of weld joints by laser welding for superplastic deformation of titanium alloy [J]. Rare Metal Materials and Engineering, 2012, 41(2): 368 371. (in Chinese) [10] CHENG Dong-hai, HUANG Ji-hua, LIN Hai-fan, ZHAO Xing-ke, ZHANG Hua. Superplastic deformation behavior and microstructures of laser welded titanium alloy [J]. The Chinese Journal of Nonferrous Metals, 2010, 20(1): 67 71. (in Chinese) [11] WEN Jiu-ba, YANG Yun-lin, YANG Yong-shun, CHEN Fu-xiao, ZHANG Ke-ke, ZHANG Yao-zong. Application of superplasticity technology [M]. Beijing: China Machine Press, 2005: 1 2. (in Chinese) [12] MULLINS W W, SEKERKA R F. Stability of a planar interface during solidification of a dilute binary alloy [J]. Journal of Applied Physics, 1964, 35(2): 444 451. [13] BISWAS S, REDDY G M, MOHANDAS T, MURTHY V S. Residual stresses in Inconel 718 electron beam welds [J]. Journal of Material Science, 2004, 39: 6813 6815. [14] RADHAKRISHNA C H, PRASAD R K. The formation and control of Laves phase in superalloy 718 welds [J]. Journal of Material Science, 1997, 32(8):1977 1984. [15] GUO Jian-ting. Materials science and engineering for superalloys(I) [M]. Beijing: Science Press, 2008: 353 360. (in Chinese) [16] LEE S B, LIAW P K, LIU C T, CHOU Y T. Cracking in Cr Cr 2 Nb eutectic alloys due to thermal stresses [J].Materials Science and Engineering A,1999, 268(1 2): 184 192. ä�™ã’šá±Š Inconel 718ড়䞥▔Ü�á‡�᥹ᵓⱘ催â�½á¢�Ԍᗻ㛑 ௗ׬ಙ 1 ēঞ༜ڛ 1 ē໺ ׆ 1 ē჆ף࢚ 2 1. 䞡ᑚ໻ᄺ á´¤á⾥ᄺϢᎹ⿟ᄺ䰶ˈ䞡ᑚ 400044; 2. જᇨⒼᎹϮ໻ᄺ á´¤á⾥ᄺϢᎹ⿟ᄺ䰶ˈજᇨⒼ 150001 ᨬ 㽕˖ЎՓ Inconel 718ড়䞥ⱘㄦᔶ໮ሖà¼�㢃㒧ᵘ乎߽៤ᔶˈⷨã�Šä†¹à§œäž¥â–”Ü�á‡�᥹ᵓⱘ催â�½à¸¥á—»Ç„㒧ᵰ㸼ᯢˈá¢� ÔŒá®�৥á‡�â–”Ü�á‡�᥹ᵓⱘԌä“⥛á³�ᕜ໻ᕅડDŽ῾৥á¢�Ԍᯊˈ೼â�½á‘ºÐŽ 950 °C੠ᑨব⥛Ў 3.1×10 4 s 1 ⱘᴵӊϟˈ᳔ à»»ÔŒä“⥛Ў 458.56%ˈℸᯊ mØ�ÐŽ 0.352ˈ⛞㓱᳾বᔶDŽ㒉৥á¢�Ԍᯊˈ೼â�½á‘ºÐŽ 965 °C੠ᑨব䗳⥛Ў 6.2×10 4 s 1 ⱘᴵӊϟˈ᳔໻ᓊä“⥛Ў 178.96%ˈℸᯊ m Ø�ÐŽ 0.261ˈ⛞㓱䱣↡ᴤৠᯊবᔶˈ⛞㓱ⱘᖂ㾖㒘㒛Ўᷥᵱ᱊ˈ᱊䯈 ᵤߎњ Nb ৿䞣䕗催ⱘ Laves ⳌDŽ㒉৥á¢�Ԍᯊˈ⬅Ѣࡼᗕݡ㒧᱊ⱘ㓬ᬙˈ⛞㓱Ð�ߎ⦄њ⬅ᷥᵱ᱊੠ã„�䕈᱊㒘៤ⱘ â�‹à§œã’˜ã’›Ç„催â�½à¦¬á”¶à§¢Ëˆâ›žã“±Ð�ⱘ໻䞣 LavesⳌ䕀࣪Ў ⳌˈԚ⛞㓱Ð�Ò¡á³�ᇣ䚼ߚ⅟Ôⱘ LavesⳌᄬ೼DŽ໮ሖà¼� 㢃ㄦ㒧ᵘ៤ᔶⱘ㒧ᵰ㸼ᯢˈ▔Ü�á‡�⛞ᵓⱘ催â�½à¸¥á—»ã›‘⒵䎇݊ㄦᔶà¼�㢃㒧ᵘⱘ㽕∖DŽ ݇䬂䆡˖Inconel 718ড়䞥˗▔Ü�⛞᥹˗催â�½à¸¥á—»Ë—ᖂ㾖㒘㒛 (Edited by HE Xue-feng) Shi-gen. Study on the microstructures and properties of GH4169 alloy by different forging technology [J]. Rare Metal Materials and Engineering, 2011, 40(S2): 225 226. (in Chinese) [4] QU Feng-sheng, LU Zhen, XING Fei, ZHANG Kai-feng. Study on LBW/SPF technology of multi-sheet cylinder sandwich structure for inconel 718 Superalloy [J]. Journal of Sichuan University: Engineering Science Edition, 2102,44(3): 185 186. (in Chinese) [5] LU Hong-jun. Study of untra-fine grain processing and superplastic forming of GH4169 superalloy sheet [D]. Harbin: Harbin Institute of Technology, 2003. (in Chinese) [6] JANAKI RAMA G.D, ENUGOPAL REDDYA A V, PRASAD RAOB K, REDDYC G M, SARIN SUNDAR J K. Microstructure and tensile properties of Inconel 718 pulsed Nd-YAG laser welds [J]. Journal of Material Processing Technology, 2005, 167: 73 82. [7] GAO Peng, ZHANG Kai-feng, ZHANG Bing-gang, JIANG Shao-song, ZHANG Bao-wei. Microstructures and high temperature mechanical properties of electron beam welded Inconel 718 superalloy thick plate [J]. Transactions of Nonferrous Metals Society of China, 2011, 21: s315 s322. [8] CHEN S H, HUANG J H, CHENG D H, ZHANG H, ZHAO X K. Superplastic deformation mechanism and mechanical behavior of a laser-welded Ti 6Al 4V alloy joint [J]. Materials Science and Engineering A, 2012, 541: 110 119. [9] CHENG Dong-hai, HUANG Ji-hua, CHEN Yi-ping, HU De-an. Microstructure evolution characterization of weld joints by laser welding for superplastic deformation of titanium alloy [J]. Rare Metal Materials and Engineering, 2012, 41(2): 368 371. (in Chinese) [10] CHENG Dong-hai, HUANG Ji-hua, LIN Hai-fan, ZHAO Xing-ke, ZHANG Hua. Superplastic deformation behavior and microstructures of laser welded titanium alloy [J]. The Chinese Journal of Nonferrous Metals, 2010, 20(1): 67 71. (in Chinese) [11] WEN Jiu-ba, YANG Yun-lin, YANG Yong-shun, CHEN Fu-xiao, ZHANG Ke-ke, ZHANG Yao-zong. Application of superplasticity technology [M]. Beijing: China Machine Press, 2005: 1 2. (in Chinese) [12] MULLINS W W, SEKERKA R F. Stability of a planar interface during solidification of a dilute binary alloy [J]. Journal of Applied Physics, 1964, 35(2): 444 451. [13] BISWAS S, REDDY G M, MOHANDAS T, MURTHY V S. Residual stresses in Inconel 718 electron beam welds [J]. Journal of Material Science, 2004, 39: 6813 6815. [14] RADHAKRISHNA C H, PRASAD R K. The formation and control of Laves phase in superalloy 718 welds [J]. Journal of Material Science, 1997, 32(8):1977 1984. [15] GUO Jian-ting. Materials science and engineering for superalloys(I) [M]. Beijing: Science Press, 2008: 353 360. (in Chinese) [16] LEE S B, LIAW P K, LIU C T, CHOU Y T. Cracking in Cr Cr 2 Nb eutectic alloys due to thermal stresses [J].Materials Science and Engineering A,1999, 268(1 2): 184 192. ä�™ã’šá±Š Inconel 718ড়䞥▔Ü�á‡�᥹ᵓⱘ催â�½á¢�Ԍᗻ㛑 ௗ׬ಙ 1 ēঞ༜ڛ 1 ē໺ ׆ 1 ē჆ף࢚ 2 1. 䞡ᑚ໻ᄺ á´¤á⾥ᄺϢᎹ⿟ᄺ䰶ˈ䞡ᑚ 400044; 2. જᇨⒼᎹϮ໻ᄺ á´¤á⾥ᄺϢᎹ⿟ᄺ䰶High temperature tensile properties of laser butt-welded plate of Inconel 718 superalloy with ultra-fine grainsFor successfully forming multi-sheet cylinder sandwich structure of Inconel 718 superalloy, high temperature tensile properties of laser butt-welded plate of Inconel 718 superalloy were studied. The experiment results show that tensile direction has great effect on elongation of the laser butt-welded plate. Under conditions of transverse direction tension, the maximum elongation reaches 458.56% at 950 °C with strain rate of 3.1×10−4 s−1, in which the strain rate sensitivity value m is 0.352 and the welding seam is not deformed. Under conditions of longitudinal direction tension, the maximum elongation is 178.96% at 965 °C with strain rate of 6.2×10−4 s−1, in which m-value is 0.261, and the welding seam contributes to the deformation with the matrix. The microstructure in as-welded fusion zone is constituted of austenite dendrites and Laves phase precipitated in interdendrites. After longitudinal direction tension, a mixed microstructure with dendrite and equiaxed crystal appears in the welding seam due to dynamic recrystallization. After high temperature deforming, many δ-phase grains are transformed from Laves phase grains but a small part of residual Laves phase grains still exist in the welding seam. The deformation result of multi-sheet cylinder sandwich structure verifies that high temperature plasticity of the laser butt-welded plate can meet the requirement of superplastic forming.Investigation of chirality and diameter effects on the Young’s modulus of carbon nanotubes using non-linear potentialsThe main goal of this research is to predict Young’s modulus of carbon nanotubes using a full non-linear finite element model. Spring elements are used to simulate molecular interactions in atomic structure of carbon nanotube. All interactions are simulated non-linearly. A parametric study is performed to investigate effects of chirality and diameter on the Young’s modulus of single walled carbon nanotubes. Unlike the results of presented linear finite element models, the results of current model imply on independency of Young’s modulus from chirality and diameter. Obtained results from this study are in a good agreement with experimental observations and published data.Carbon nanotubes (CNTs) have received considerable attraction among other nano-structured materials as the reinforcing agent of composite materials. Qian et al. The remarkable and extraordinary mechanical, thermal and electrical properties of CNTs stimulated several researches to measure their properties theoretically and/or experimentally. The experimental measurement of CNT properties is a challenging task. Extremely scattered data obtained through experimental observations have encouraged many researchers to pursue a variety of theoretical studies on the effective properties of nanotubes. The theoretical studies on the mechanical properties of CNT can be divided into atomistic modeling Atomistic modeling techniques are limited to short time and small length scales preventing them to be applicable to the large number of atoms demanding huge amount of computations Continuum mechanics-based approaches employ theories of rods, trusses, beams, shells, or curved plates The demand for improvements in the modeling techniques and the development of faster methods to compute the mechanical properties of CNTs have motivated researchers to employ the FEMs which can be also referred to as numerical nanoscale continuum mechanics approaches. Different kinds of finite elements, including rods, trusses, beams, and springs, have been used to model the carbon–carbon (C–C) link in CNTs Some researchers used FEM to simulate CNT or graphene sheet using linear interatomic potentials for C–C bonds In contrast with available finite element models taking into account linear behavior for all or some interactions in CNT, in this study a full non-linear finite element model is developed using spring elements accounting for both C–C bond stretching and C–C–C bond angle variations. A parametric study is accomplished to investigate effect of CNT diameter and chirality on its Young’s modulus.Carbon nanotubes have a lattice structure comprising of periodic hexagonal network of bonded carbon atoms and maybe the fullerene like end caps. The generation of the tubular structure can be imagined by rolling up a single graphene sheet and construction of a hollow cylinder. Atomistic structure of CNT can be defined using chiral index. specifies the type of nanotube. Vector (n, |
n) and (n, 0) represents Armchair and Zigzag nanotubes, respectively and any other form are identified chiral nanotube There are different interactions between carbon atoms in CNT structure. The motions of carbon atoms are regulated by a force field that is generated by electron–nucleus interactions and nucleus–nucleus interactions where ur, uθ, uϕ, and uw correspond with bond stretching, bond angle variation, dihedral angle torsion and out-of-plane torsion, respectively. uvdw and uel are also non-bonded energies associated with van der Waals and electrostatic interactions. Generally, the dominant parts of inter-atomic potential are bond stretching and bond angle variations due to their significant contribution comparing with other interactions De=6.03105e-19(Nm),β=2.625e10(m-1),r00.142(nm).The Morse potential for simulating C–C–C bond angle variations and required parameters are given as kθ=0.9e-18N·mrad2,ksextic=0.754(rad-4),θ0=2.094(rad)In the present structural model, bond stretching associated with C–C bonds and bond angle variations attributed to adjacent C–C–C are simulated using non-linear spring elements. Finite element model is constructed in commercial finite element package ANSYS 12 Morse potential function for C–C–C bond angle variations is expressed as below using Eq. For sake of simplicity, the bond angle variations are modeled using non-linear springs connecting the opposite atoms of the C–C–C as depicted in . This strategy was firstly developed by Odegard et al. , when the C–C–C bond angle varies, the length of corresponding “B”-spring is changed. Consequently, bond angle variations are taken into account on the basis of length variations associated with “B”-springs.For small deformations, changes in length of “B”-spring elements can be obtained from this equation where r0 is the initial length of C–C bond and ΔR is the change in the length of corresponding spring element. Consequently, force–displacement relationship for spring type “B” is obtained as:F(R-R0)=duθdR=4r02kθ(R-R0)1+48r04ksextic(R-R0)4The variations of forces against displacement for both “A” and “B” springs are shown in A built-in spring element called COMBIN39 is used in ANSYS It is observes that neglecting bond angle variations and constructing the finite element model of CNT using just bond stretching interaction will lead us to an unstable structure of CNT. Namely, applying small force or displacement to CNT structure will result in a disrupted structure. Hence the bond angle variation plays an important role in simulation of CNT using spring-based FEM. This phenomenon was also reported by Belytschko et al. The model is subjected to uniform tensile displacement in axial direction at one ends, while other ends are fully restricted from any translational and rotational movements. Non-linear analysis using full Newton–Raphson method is performed on each constructed model due to employed non-linear spring elements and resultant forces are obtained at the strains of 0.1%.Several nanotubes with different chiral indices are modeled and their Young’s moduli are obtained. The aspect ratios of all models are greater than 10 in order to avoid edge effects. Resultant forces are read from the output and following equation is used to calculate Young’s modulus accordingly:The thickness of CNT is assumed as 0.34 nm which is interlayer spacing of graphene sheets in graphite. This value has been widely used by other researchers as well , the Young’s modulus of Armchair nanotubes is only slightly (0.16–0.29%) higher than that of Zigzag one which is negligible. It can be stated that the Young’s modulus of CNTs is independent from their chirality. It can also be seen from that variations of the Young’s modulus of CNTs in term of diameter changes are also very small. These variations are in the range of 1.5–3.5% and 1.8–4.9% for Armchair and Zigzag CNTs, respectively. In other words, the Young’s modulus of CNT is independent from its diameter, specifically in large CNT diameters. This behavior was also reported by other researchers The main reason of difference between linear models and non-linear ones is attributed to the curvature effects in nano-structure of CNT. In other word, linear models overestimate the effect of curvature due to the employed harmonic potentials resulting in lower values for Young’s modulus in comparison with non-linear models. However, obtained results from non-linear models The elastic modulus of SWCNTs asymptotically converges to the Young’s modulus of graphene sheet at very large diameters of CNT. The main reason for this behavior is originated from the higher curvatures of CNTs with small diameters. This will provide greater distortion in C–C bonds in smaller diameter of CNT which are much more significant than it in larger diameters. As the nanotube diameter increases, the influence of curvature diminishes gradually , the linear model of Giannopoulos et al. The linear models report that the Young’s modulus of Zigzag CNT is smaller than that of Armchair one Obtained results from this study are in a good agreement with experimental observations and published data by other researchers in literature. A comparison is inserted in The differences between reported values for Young’s modulus of CNT by different researchers in are arisen from different employed approaches and also different interatomic potentials to simulate interactions between carbon atoms. that using non-linear models will address higher values of CNT Young’s modulus (about 1–1.325 TPa) in comparison with obtained results by linear models. Some other researchers reported this value for the Young’s modulus of CNTs in literature In the present study, a fully non-linear finite element model is developed to simulate CNT using spring element. Both bond stretching and bong angle variations are considered as non-linear interactions. The importance of considering C–C–C bond angle variations is discussed and it is shown that neglecting this interaction will cause an unstable model in spring-based FEMs.The developed model is just in need of force–displacement relationships characterization to simulate molecular interactions. Consequently, one of the important advantages of this model is its simplicity. On the other hand, spring element can capture the real behavior of C–C bonds more accurately than other elements, since C–C bonds stay straight and will not bend.A graphene sheet and several Armchair and Zigzag CNTs are modeled using developed non-linear FEM. Applying uniaxial tension to the models, the Young’s moduli of the models are obtained and compared with other developed models and experimental observations in literature. A sensitivity analysis is conducted to investigate influence of CNT diameter and chirality on its Young’s modulus. It is observed that the Young’s modulus of CNT is fully independent from chirality and it is almost independent from the diameter. It is found out that obtained Young’s modulus of CNT on the basis of non-linear interatomic potentials are higher than that of linear models.Cationic and anionic composition dependence of elastic propertiesCationic and anionic concentration dependent elastic properties of zinc blende specimens within CdxZn1-xSySe1-y quaternary system: Calculations with density functional theoryElastic properties of zinc blende CdxZn1-xSySe1-y quaternary alloys have been calculated with density functional theory oriented FP-LAPW scheme. Elastic stiffness constants and hardness of specimens have been increased almost linearly with increasing sulphur composition at each fixed cadmium composition, while their decrease have been observed with increasing cadmium composition at each fixed sulphur composition in any binary-ternary/ternary-quaternary system. Mechanical and dynamical stability, elastic anisotropy, compressibility, ductility and fair plasticity have been calculated as the key features of each specimen. Mixture of covalent and ionic bonding with dominant role of covalent, central nature of interatomic forces and bending over stretching in chemical bonds have been observed in each compound. Calculated Debye temperature suggests ZnS as the stiffest and CdSe as the softest among all compounds. Calculated Gruneisen parameter has confirmed anharmonic nature of interaction between the atoms. Thermal conductivities and melting temperatures of all the specimens have also been computed.Cationic and anionic composition dependence of elastic propertiesFabrication of quaternary alloys has been considered as an advanced procedure for manipulating various properties with improved accuracy compared to the ternary alloys in materials science and it expands the range of their target-oriented applications. The proper choice of constituent binary compounds plays the key role behind the success of such fabrication procedure. In this paper, elastic properties of zinc blende specimens under CdxZn1−xSySe1−y system, bounded by two anionic ternary CdSySe1−y, ZnSySe1−y and two cationic ternary CdxZn1−xS, CdxZn1−xSe systems, have been reported. All the ternary compounds have again been originated from the four basic binaries CdS, CdSe, ZnS and ZnSe. Therefore, elastic properties of such ternary and quaternary alloys would be completely different from their basic binary constituents.Group IIB–VIA diatomic wide-direct-band-gap (Γ-Γ) cadmium chalcogenides CdS and CdSe and zinc chalcogenides ZnS and ZnSe semiconductors are available in zinc blende (B3) phase [] under ambient conditions. Visual displays, laser diodes operating in blue-green spectral range, hetero-structure lasers having wide band gap, photovoltaic devices, optical wave guides and memories, thin-film transistors, UV-light sensors, chemical sensors, biosensors and different types of nanostructures are widely prepared with these diatomic semiconductors [The elastic features of a solid are linked to its fundamental properties like phonon spectra, inter-atomic potentials, equation of state, specific heat, Debye temperature, melting point, thermal expansion etc. Several experimental investigations on elastic and related properties of cadmium/zinc chalcogenides have been carried out []. From the measurements of ultrasonic wave velocities, elastic moduli of hexagonal CdSe at 25 °C, adiabatic bulk modulus, volume compressibility and Debye temperatures have been determined by Cline and coworkers []. Elastic constants of wurtzite CdS and CdSe as well as cubic ZnS and ZnSe were measured by Berlincourt []. The pressure dependences of elastic constants of zinc blende ZnSe in single crystals at 770–3000 K [] were investigated using ultrasonic pulse-echo method by Lee. Wave shock induced phase transitions and Hugoniot elastic limits for ZnS and ZnSe were investigated by Gust []. Elastic constants of ZnSe at 2950 K and its Brillouin frequency were measured from Brillouin scattering experiments by Hodgins and Irwin []. Phonon density of states and Debye temperature for cubic ZnS were measured from neutron scattering experiments by Vagelatos []. Experimentally measured Phillip ionicities of chemical bonds in zinc blende CdS, ZnS and ZnSe crystals were reported by Phillips []. Further, some of experimentally investigated optoelectronic and elastic properties of zinc and cadmium chalcogenides have also been reported in a couple of literatures [Structural and elastic features of diatomic cadmium and zinc chalcogenides were also reported from several density functional investigations. Chen and Mintz have computed lattice constants for diatomic cadmium and zinc chalcogenides employing Gaussian dual-space density functional theory []. Also, lattice constants of zinc and cadmium chalcogenides have been calculated from some first principles investigations []. A simple phenomenological theory on the elastic properties of zinc blende semiconductors was introduced for the first time by Martin []. Kamran and coworkers have calculated the shear modulus and bulk modulus for zinc blende cadmium and zinc chalcogenides with the semi-empirical formula, developed by themselves for covalent compounds []. Using a three-body force potential approach, Singh and Singh have calculated the pressure dependence of second and third order elastic properties of ZnS, ZnSe and ZnTe []. Khenata and coworkers have carried out density functional full-potential linearized augmented plane wave (FP-LAPW) calculations of elastic features of ZnS, ZnSe and ZnTe under applied pressure []. The elastic shear constants, internal-strain parameters and deformation potentials for cubic ZnS and ZnSe under hydrostatic pressure were calculated by Casali and Christensen with FP-LAPW approach []. Bilal and coworkers have calculated elastic properties of cubic Zn-chalcogenides under hydrostatic pressure with FP-LAPW approach []. Guo and coworkers have calculated structural and elastic properties of CdS, CdSe and CdTe using pseudopotential plane wave (PP-PW) method []. Density functional calculations on elastic properties of CdS, CdSe and CdTe were performed by Deligoz and coworkers []. Elastic properties of CdS, CdSe and CdTe were calculated by Sharma and coworkers from first principle calculations []. Elastic constants for zinc blende CdS were reported by Wright and Gale [] from calculations of respective interatomic pair potentials. Elastic shear constants and Phillip ionicities of ZnS, ZnSe and CdS were reported by Kitamura and coworkers employing extended Huckel theory []. Ouendadji and coworkers have calculated structural, elastic and thermal properties of CdS, CdSe and CdTe with FP-LAPW scheme [Despite of large number of experimental and theoretical studies on elastic features of diatomic CdS and CdSe as well as ZnS and ZnSe, we cannot find any kind of study on elastic features of any of the anionic ternary CdSySe1−y and ZnSySe1−y alloys, cationic ternary CdxZn1−xS and CdxZn1−xSe alloys and CdxZn1−xSySe1−y quaternary alloys. These facts motivate us to perform detailed theoretical studies on elastic properties of all the zinc blende ternary and quaternary alloys under CdxZn1−xSySe1−y systems for the first time along with those of their constituent binary compounds for the entire cationic (Cd) composition x and anionic (S) composition y range 0.0 ≤ x, y ≤ 1.0. Moreover, segregating the entire CdxZn1−xSySe1−y quaternary system into five binary-ternary/ternary-quaternary sub-systems, studies on anionic (sulphur) composition (y) and cationic (cadmium) composition (x) dependences of elastic properties of the specimens in these systems have been carried out in an elaborate way.] has been employed practically with WIEN2K code [], which can successfully explores various features of solids. The cubic2-elastic package of Jamal and coworkers [], accessory of WIEN2K, has been used to compute mechanical properties of our considered zinc blende specimens. The Perdew-Burke-Ernzerhof generalized gradiant approximation (PBE-GGA) scheme [] has been employed to compute exchange-correlation (XC) potentials for the said properties.Initially, the zinc blende (F4‾3m) unit cells of ZnS, ZnSe, CdS and CdSe have been simulated considering their respective experimental lattice constants []. The cationic ternary CdxZn1-xS and CdxZn1-xSe unit cells for x = 0.25, 0.50 and 0.75 have been simulated by replacing zinc with cadmium atom(s) gradually in the ZnS and ZnSe unit cells, respectively, while anionic ternary ZnSySe1-y and CdSySe1-y unit cells for y = 0.25, 0.50 and 0.75 have been simulated by replacing selenium with sulphur atom(s) gradually in the ZnSe and CdSe unit cells, respectively. For each of the fixed x = 0.25, 0.50 and 0.75, the CdxZn1−xSySe1−y quaternary unit cells have been simulated for y = 0.25, 0.50 and 0.75 by gradually replacing selenium with sulphur atom(s) from the respective unit cell of CdxZn1-xSe. There are eight atoms in each of the simulated zinc blende binary, ternary and quaternary unit cells, which have been visualized by utilizing XCrySDen [The muffing-tin model has been adopted in the FP-LAPW approach. Only non-overlapping muffin-tin spheres contain atoms, where spherical harmonics expansion of the Kohn–Sham wave functions have been considered with maximum angular momentum lmax = 10. Inside interstitial regions of the muffin-tin structure, we have considered the expansion of Kohn-Sham wave functions with plane wave basis taking into account the Kmax = 8.0/RMT as the largest K-vector and Gmax = 16 Ry1/2 as the Fourier expansion parameter. In order to fulfill the criteria of non-leakage of charge from the core of atomic spheres, RMT = 2.5, 2.4, 2.2 and 2.3 a. u. have been considered as the smallest radius of muffin-tin spheres containing Cd, Zn, S and Se atoms, respectively. We have carried out the integrations over the first Brillouin zone considering a mesh of 1500 k-points. Self-consistent-field (SCF) technique with a threshold 10−5 Ry has been employed for achieving the convergence in total energy.Minimum total energy of each simulated unit cell is calculated self-consistently with respect to cell parameters and atomic positions and hence equilibrium lattice constant (a0) of any cubic unit cell has been calculated by fitting the total cell energy verses volume curve to Murnaghan's equation of state []. Calculated a0 for all types of specimens have been reported in . Calculated a0 for each binary compound agrees well with the corresponding experimental a0 for respective zinc and cadmium chalcogenides []. Moreover, our calculated lattice constant a0 for each binary chalcogenide also agrees well with respective previously calculated data for ZnS and ZnSe []). We can not compare calculated a0 for any of the ternary or quaternary specimens due to lacking of experimentally investigated or previously calculated data. (a) shows that in any binary-ternary/ternary-quaternary system, calculated lattice constant a0 has been decreased nonlinearly with increasing sulphur composition y at each cadmium composition x. (a) also states that a0 has been increased with increasing x at each y.In a crystal, number of required deformations and hence number of independent elastic constants decreases due to enhancement in number of symmetries and vice versa []. A cubic crystal system provides the simplest form of elastic matrix due to high symmetry. It provides only non-zero independent elastic stiffness constants C11 related to longitudinal elastic behaviour as well as C12 and C44 related to shearing deformation in the crystal [The calculated elastic stiffness constants for each of the binary, ternary and quaternary specimens under consideration have been reported in (b–d) have shown almost linear increase in calculated C11, C12 and C44, respectively, with increasing sulphur composition y at each cadmium composition x in any binary-ternary/ternary-quaternary system. In contrary, decrease in each of the C11, C12 and C44 with increasing x has been observed at each y.Experimental elastic stiffness constants for CdS, CdSe, ternary and quaternary alloys are not available. In contrary, calculated C11, C12 and C44 for ZnS and ZnSe are very close to the corresponding experimental data []. Also fair agreement between our calculated and some respective previously calculated C11, C12 and C44 have been observed for the binary compounds in , the trend C11>C12>C44 for each of the binary, ternary and quaternary specimens have been observed. Such trend for only the binary compounds has been verified by experimental [] or some previous theoretical studies [ shows that Born's stability criteria C11−C12>0; C11+2C12>0; C11>0; C44>0[]; have been fulfilled and hence mechanical stability has been exhibited by each cubic specimen. Computed shear constant C/=(C11−C12)/2 of each of the considered specimens is also presented in . Due to calculated positive C/, each compound under consideration shows dynamical stability []. Calculated C/ has been increased with increasing sulphur composition y at each cadmium composition x, while it has been decreased with increasing x at each y in any binary-ternary/ternary-quaternary system. For any compound under consideration, no experimental C/ is available for comparison, while that for CdS is substantially higher than a corresponding previously calculated data [The shear modulus indicates response and hence the resistance of solid to the shearing deformation. The upper limit of effective shear modulus is represented with Voigt (GV) [], while the lower limit with Reuss (GR) [] shear modulus. The GV and GR for a cubic specimen have been expressed with C11, C12 and C44 as [The intermediate conditions for a cubic crystal is indicated with Hill's shear modulus (GH) [] through either GHA=(GV+GR)/2 or GHG=GVGR []. Calculated GV,GR,GHA and GHG for all the compounds have been presented in . Calculated GHA and GHG of each of the specimens have been found to be very close to each other. (a–d) have shown almost linear increase in each of the calculated GV, GR, GHA and GHG, respectively, with increasing sulphur composition y at each cadmium composition x in any binary-ternary/ternary-quaternary system. In contrast, decrease in each of the calculated GV, GR, GHA and GHG with increasing x has been observed at each y from (a–d). Calculated GHA for CdS and CdSe are fairly larger than the respective existing experimental GHA for CdS [] as well as some of their previous theoretical GHA [The upper (GU) and lower (GL) boundaries of shear modulus were alternately proposed by Hashin and Shtrikman [GU=G2+2[5G1−G2+18(B0+2G2)5G2(3B0+4G2)]−1GL=G1+3[5G2−G1+12(B0+2G1)5G1(3B0+4G1)]−1Here, G1=(C11−C12)/2 and G2=C44. The intermediate shear modulus is known as Hashin-Shtrikman shear modulus, expressed with the average of GU and GL as GHS=(GU+GL)/2 []. The calculated GU, GL and GHS for all the compounds have been reported in (a–c) have shown almost linear increase in calculated GU, GL and GHS, respectively, with increasing sulphur composition y at each cadmium composition x in any binary-ternary/ternary-quaternary system. In contrast, decrease in each of the calculated GU, GL and GHS with increasing x has been observed at each y from (a–c). No experimental or previously calculated GU, GL and GHS are available for any of the specimens under consideration.The bulk modulus (B0) of a solid measures the opposition of the material to iso-static contraction or tensile stress acting uniformly over its surface and it is reciprocal to its compressibility. On the other hand, Young's modulus (Y) is the signature of rigidity in any solid under uniaxial deformation. Improved hardness of a solid is indicated with its larger B0 or Y and vice versa. In a cubic crystal, B0 has been expressed with C11 and C12, while Y with B0 and GHA in the following way [The hardness of a solid can also be calculated with its Vicker's hardness (HV). In terms of Hill shear modulus (GHA), the HV is expressed with two different empirical relations. Teter [] has proposed the first one as HVT=0.1769GHA−2.899, while Chen and coworkers [] has proposed the second one as HVC=0.151GHA.The calculated B0, Y, HVT and HVC for the specimens under consideration have been reported in . The conditionC12<B0<C11, fulfilled by each specimen, again confirms the respective mechanical stability. No experimental data for Y as well as both experimental and previous theoreticl HVT and HVC for any of the binary, ternary and quaternary alloys are available. Excellent agreement between calculated B0 for ZnS, ZnSe and CdS and the respective experimental [] and some of the previously calculated [] data has been observed. Calculated B0 for CdSe is fairly smaller than the corresponding experimental [] data. In case of Y, the only previously calculated data for CdS [] are in fair agreement with our respective calculated data. (a–d) have shown almost linear increase in calculated hardness of specimens in terms of each of the B0, Y, HVT and HVC, respectively, with increasing sulphur composition y at each cadmium composition x in any binary-ternary/ternary-quaternary system. In contrast, decrease in each of the calculated B0, Y, HVT and HVC with increasing x has been observed at each y from (a–d). Calculations of B0, Y, HVT and HVC indicate that ZnS and CdSe has the maximum and minimum hardness, respectively.The plasticity in a material is determined by Ref. B0/C44 [] and the same for the specimens under consideration have been presented in . Calculated B0/C44 for each of the specimens, greater than unity, indicate their fair plasticity. (a) has shown that in any binary-ternary/ternary-quaternary system, calculated B0/C44 increases nonlinearly with increasing sulphur composition y at each cadmium composition x. Also, increase in B0/C44 with increasing x has been observed at each y from (a). Experimental or previously calculated B0/C44 data for any specimen under consideration is unavailable.] are the signatures of ductility or fragileness of a solid. The threshold B0/GHA = 1.75 is acting as a separator between ductile and fragile character of materials. The B0/GHA, higher and lower than the threshold, stand for ductility and brittleness, respectively, of a solid. Also, positive and negative sign of C// indicates ductility and brittleness, respectively, of a material. Our calculated B0/GHA>2.0 and positive C// for each specimen, reported in , indicate ductile nature of each specimen. Experimental B0/GHA for the specimens under consideration are unavailable. Our calculated B0/GHA for CdS and CdSe are fairly smaller than the reported corresponding previously calculated data [(b) has shown that calculated B0/GHA increases nonlinearly with increasing sulphur composition y at each cadmium composition x in any binary-ternary/ternary-quaternary system. Also, increase in B0/GHA with increasing x has been observed at each y from (b). Experimental or earlier calculated C// for the any specimen under consideration is unavailable in literature. , presenting the correlations between C// and B0/GHA of all the compounds, confirms that all the considered specimens exist in the ductile region.Different aspects of a material can be explored by calculating its Poisson's ratio σ (−1≤σ≤0.5). The tendency of total incompressibility of the material is expressed by σ→0.5, central type of bonding force by 0.25<σ<0.50, better plasticity by σ closer to 0.5 and poor plasticity by σ closer to −1.0, brittleness by σ<0.26 and ductility by σ>0.26 []. Poisson's ratio σ of any material can be expressed with B0 and GHA as [, calculated σall the considered compounds have been reported. The calculated σ in the range 0.29–0.33 for all the compounds states that interatomic bonding force in each of them is a central force. Moreover, calculated σ in the aforesaid range shows considerable compressibility, ductility and plasticity. (a) has indicated that calculated σ increases nonlinearly with increasing sulphur composition y at each cadmium composition x in any binary-ternary/ternary-quaternary system. Also, increase in σ with increasing x has been observed at each y from (a). Experimental Poisson's ratio (σ) for any specimen is unavailable, while calculated σ for CdS and CdSe agree fairly well with respective earlier calculated σ for CdS and CdSe [Under strain distortions at conserved volume, the relative positions of anionic and cationic sublattices in a material have been explained with its Kleinman parameter ξ (0≤ξ≤1) []. The ξ = 0 is indicating minimized bending, while ξ = 1 is indicating minimized stretching of bonds. The ξ of a cubic crystal in terms of C11 and C12 has been expressed as [Calculated ξ for each of the considered specimens has been presented in . Our calculated ξ in the range 0.70–0.76 indicates significantly dominating bending of bond over stretching in each specimen under consideration. (b) has indicated that calculated ξ increases nonlinearly with increasing sulphur composition y at each cadmium composition x in any binary-ternary/ternary-quaternary system. Also, increase in ξ with increasing x has been observed at each y from (b). Experimental ξof any specimen is unavailable, while calculated ξ of CdS and CdSe are marginally smaller than their respective previously calculated data [The elastic anisotropy is a signature of directional dependence of bonding nature in a crystal, which is responsible for possibility of existence of microcracks in that crystal. It is generally calculated with Zener anisotropy factor (Az). The Az for a cubic crystal has been expressed with C11, C12 and C44 in the following way [The Az = 1 stands for 100% elastic isotropy. In contrast, Az>1 indicates anisotropy with maximum rigidity along <111> cube diagonal, while that along <100> cube axis by Az<1 Ref. [Alternately, elastic anisotropy in a crystal can be examined in two different ways. The following way of calculating percentage of anisotropy (AG) in a crystal in terms its GV and GR was proposed by Chung and Buessem [Elastic anisotropy in a cubic crystal has also been explored by computing the anisotropy in elastic wave velocity (Ae) in the following manner [In any crystal, AG=0 and AG = 1 stand for 100% isotropy and anisotropy, respectively. Though Ae = 0 fits for perfect isotropy, the elastic anisotropy has been indicated with its departure from zero.Calculated Az, AG and Ae for all the specimens have been presented in . Calculated Az>2 for each of the specimens points toward elastic anisotropy with maximum rigidity along <111> cube diagonal. Again, calculated AG and Ae for all the specimens in the ranges 0.082–0.098 and 0.864–0.947, respectively, again indicate their elastically anisotropic nature. (a–c) have indicated nonlinear increase in calculated Az and AG, but decrease in Ae with increasing sulphur composition y at each cadmium composition x in any binary-ternary/ternary-quaternary system. Also, marginal increase in Az and AG, but decrease in Ae with increasing x have been observed at each y from (a–c), respectively. No experimental Az as well as experimental or previous theoretical AG and Ae for any of the specimens is available. Our calculated Az for CdS and CdSe lie between a couple of respective previously calculated data [For a cubic crystal, Lame's constants λ and μ with the conditions λ=C12, and μ=C/ = C44 determine the elastic isotropy, while violation of these conditions indicate elastic anisotropy []. The λ and μ have been expressed with Y and σ in the following way [Calculated λ and μ for the specimens under consideration have been presented in , where a clear breaching of the aforesaid conditions is indicating their elastically anisotropic nature. In any binary-ternary/ternary-quaternary system, calculated λ and μ have been increased with increasing sulphur composition y at each cadmium composition x, while the reverses have been observed with increasing x at each y from . No experimental or previously calculated λ and μ for any specimen is available to compare.] proposed that in any zinc blende crystal, elastic constant ratio C11/C12 = 1.0, C11/C12 = 2.0 and 1.0< C11/C12< 2.0 indicate purely ionic, purely covalent and mixed character of bonding, respectively.] proposed a relation between the elastic constant ratio C11/C12 and the ionic charge Z/Z0 in the following wayClearly, increase in C11/C12 results decrease in Z/Z0 and vice versa is indicating Z/Z0 = 1.0 for completely ionic, Z/Z0 = 0.155 for completely covalent and intermediate value for mixed type of bonding in any crystal., calculated C11/C12 and Z/Z0 for all the zinc blende specimens have been reported. Clearly, calculated C11/C12 and Z/Z0, lying in the range 1.526–1.687 and 0.340–0.464, respectively, indicates mixed character of bonding with dominating role of covalent character over ionic in each crystal. In any binary-ternary/ternary-quaternary system, (a) and (b) have indicated nonlinear decrease in calculated C11/C12, but increase in Z/Z0, respectively, with increasing sulphur composition y at each cadmium composition x. Also, decrease in C11/C12 as well as increase in Z/Z0 with increasing x has been observed at each y from (a) and (b), respectively. No experimental or previously calculated C11/C12 and Z/Z0 for any specimen is available to compare.Relations between covalency αC and elastic stiffness constants C11, C12 and C11/C12 have been expressed on the basis of bond-orbital model as [C11=B0(1+αC2),C12=B0(2−αC22)andC12C11=(2−αC22+2αC2)In any zinc blende crystal, Philip ionicity fi has been calculated with the following relation [, we have reported the calculated αC and fi of each zinc blende specimen. In any binary-ternary/ternary-quaternary system, (a) and (b) have indicated nonlinear decrease in calculated αC, but increase in fi with increasing sulphur composition y at each cadmium composition x. Also, decrease in αC, but increase in fi with increasing x have been observed at each y from (a) and (b), respectively. Experimental or previous theoretical αC for any specimen is unavailable, while very good accordance between our calculated fi for CdS and CdSe and the corresponding experimental data [The Debye temperature (θD) of a solid is the approximate characteristics temperature limit distinguishing the quantum mechanical and classical characteristics of phonons in any solid. All vibrational modes have constant energy KBT in the temperature regime T >θD. In contrary, the solid exhibits quantum mechanical behaviour and high-frequency modes of vibration are kept detained at T<θD. Therefore, all the vibrational excitations completely arise from acoustic vibrations at T<θD and hence Debye temperature θD of a solid can be calculated from its elastic constants. According to a standard model, it can be expressed in terms of average sound velocity (vm) []. The mean average sound velocity vm of a specimen has been expressed as [The longitudinal velocity (vl) and transverse velocity (vt) of acoustic wave have been expressed in terms of calculated GHA and B0 as [Calculated vl, vt and vm for each of the specimens under consideration have been reported in . In any binary-ternary/ternary-quaternary system, (a–c) have indicated nonlinear increase in calculated vl, vt and vm, respectively, with increasing sulphur composition y at any cadmium composition x. In contrast, decrease in each of the vl, vt and vm with increasing x has been observed at each y from (a–c), respectively. No experimental or previous theoretical acoustic velocities vl, vt and vm of any specimen are available for comparison.In terms of mean acoustic wave velocity (vm), Debye temperature of a solid has been expressed in the following way [Here, n, ρ,M, Na,h, KB stand for number of atoms in the unit cell, density of the material, molecular weight of the unit cell, Avogadro number, Planck's constant and Boltzmann constant, respectively.During thermal transport, frequency of maximum atomic vibrations in a solid has been termed as Debye frequency (ωD). In case of a solid, it has been calculated with Debye temperature (θD) using the following relation [Calculated θD and ωD for each of the specimens under consideration have been reported in . In any binary-ternary/ternary-quaternary system, (a) and (b) have indicated that calculated θD and ωD, respectively, increases nonlinearly with increasing sulphur composition y at each cadmium composition x. In contrast, decrease in each of the θD and ωD with increasing x has been observed at each y from (a) and (b), respectively. No experimental θD and ωD of any specimen is available for comparison. Only reported θD for CdS and CdSe from a previous theoretical calculation [] are substantially lower than our respective calculated data. Enhanced Debye temperature θD in a solid is a signature of stronger inter-atomic force and hence stiffness of the solid and vice versa []. Therefore, our calculations have proposed ZnS as the stiffest and CdSe is the softest among all the compounds under consideration.The Gruneisen parameter (γa) of a solid is connected to anharmonicity in inter-atomic interactions, through which nonlinear acoustical properties are linked to its elastic nonlinearity. The dependence of elastic properties of a solid on temperature, thermal expansion and conduction of a solid, absorption of acoustic waves in a solid etc. are exclusively controlled by its γa []. Calculation of γa for a solid indicates volume dependence of lattice vibrations and temperature dependence of size or dynamics of crystal lattice in a solid. It has been obtained from the vl and vt of a solid in the following way [Calculated γa for all the specimens under consideration, presented in , confirm anharmonicity in interaction between the atoms in all the cubic specimens. In any binary-ternary/ternary-quaternary system, (c) has indicated nonlinear increase in calculated γa with increasing sulphur composition y at each cadmium composition x. Also, increase in γa with increasing x has been observed at each y from (c). We are unable to compare calculated γa for any of the considered specimens due to lacking of experimentally investigated or previously calculated data.The heat conduction capability of any solid is indicated by its thermal conductivity. It has been recognized as very important factor for applications of a material in thermal barrier coatings [] have proposed an empirical relation for estimating minimum thermal conductivity (Kmin) of a solid asAn empirical formula was proposed by Fine and coworkers [] for calculating melting temperature (Tm) of a zinc blende specimen. In terms of elastic constant C11, the Tm has been expressed as, calculated Kmin(W.m−1.K−1) and Tm (K) for all the considered have been reported. In any binary-ternary/ternary-quaternary system, (a) and (b) have indicated nonlinear increase in calculated Kmin and almost linear increase in calculated Tm, respectively, with increasing sulphur composition y at each cadmium composition x. In contrast, decrease in Kmin and Tm with increasing x have been observed at each y from (a) and (b), respectively. We are unable to compare calculated Kmin and Tm for any of the considered specimens due to lacking of experimentally investigated or previously calculated data.The mechanical aspects of zinc blende binary, ternary and quaternary specimens under CdxZn1-xSySe1-y quaternary system have been explored with density functional theory oriented FP-LAPW scheme. Elastic stiffness constants as well as hardness of specimens in terms of Young's modulus, shear modulus and Vicker's hardness have been increased almost linearly with increasing sulphur composition at each cadmium composition, while reverses have been observed with increasing cadmium composition at each sulphur composition in any binary-ternary/ternary-quaternary system. Calculated elastic stiffness constants and shear constant indicate mechanical and dynamical stability of each specimen. Computed bulk modulus, Young's modulus, Shear modulus and Vicker's empirical hardness parameters for each specimen provides idea about hardness of respective specimen. Calculated Zener anisotropy and Lame's constants have indicated that each specimen is elastically anisotropic in nature. Significantly prominent bending over stretching of bonds has been explored with calculated Kleinmann parameter. Ductile behaviour of any specimen has been explored from respective calculated Poisson's ratio, Pugh's ratio and Cauchy pressure. Calculated Poisson's ratio and B0/C44 indicate fair plasticity of each specimen under consideration. Calculated C11/C12, Z/Z0, αC and fi have indicated mixture of covalent and ionic bonding in any specimen with vital role played by the covalent nature. With increasing sulphur composition at each cadmium composition, we have observed nonlinear increase in each of the calculated longitudinal, transverse and mean sound velocity, Debye temperature and frequency, thermal conductivity and almost linear increase in melting temperature, while reverses have been observed with increasing cadmium composition at each sulphur composition in any binary-ternary/ternary-quaternary system. Calculated Debye temperature suggests ZnS as the hardest and CdSe is the softest among all the compounds under consideration. Calculated Gruneisen parameter, showing anharmonic type of interactions between atoms, has been used in investigating thermo-elastic stress of materials and it has been increased nonlinearly in both types of variations in any binary-ternary/ternary-quaternary system. The possibilities of application of considered specimens in thermal barrier coatings and different thermal management systems have been indicated with respective calculated thermal conductivity and melting temperature.The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.Structural, mechanical and corrosion evaluations of Cu/Zn/Al multilayered composites subjected to CARB processIn this research, the structure, tensile, wear, fracture toughness, and corrosion characteristics of Cu/Zn/Al multilayered composites fabricated by cross accumulative roll bonding (CARB) were determined. X-ray diffraction and scanning electron microscopy were used for structural analyses. Results showed that the crystallite size of the Cu matrix is reduced to about 50 nm at the ninth cycle of CARB, while the plastic instability and shear bands appeared in the layers after the third cycle. The tensile strength of the composites increased up to the third CARB cycle and then decreased to the ninth CARB cycle gradually. Typically, the maximum strength of about 330 MPa and elongation of almost 31.5% were obtained at the third and first cycles, respectively. A similar trend was found for the fracture toughness of the samples, so that the maximum fracture toughness of about 31 MPa.m1/2 was obtained at the third cycle. It was also shown that the wear mechanism of the composite samples changes by increasing the cycles, a compromise of abrasion and delamination wear. The accommodation of Al and Zn layers toward the surface of the composites by increasing the CARB cycles was also recognized to be responsible for the decrease of the corrosion resistance determined by polarization experiments.One of the most effective and economical ways to improve mechanical properties of materials is fabrication of metal matrix multilayered composites accompanied by refining microstructure Severe plastic deformation is a process that produces ultrafine-grained materials by applying intense plastic strain to the material below the recrystallization temperature without intermediate heat treatment So far, no systematic study has been reported on the fracture toughness and corrosion behavior of Cu/Zn/Al composites produced by the CARB process. In the present study, Cu/Zn/Al multilayered composites were fabricated at various cycles of CARB at room temperature. The structure, fracture toughness and corrosion behavior of the produced composites were examined.The raw materials used in this study were commercially pure Al (1100-Al alloy), Cu, and Zn sheets with the primary dimensions and mechanical properties listed in schematically shows the CARB fabrication process which consists of two steps. Fabrication of the initial sandwich at the first cycle named the first step. To produce the initial sandwich, three Cu, two Al, and two Zn sheets were cut at the same dimensions and were stacked on each other by the Cu/Zn/Al/Cu/Zn/Al/Cu sequence after surface preparation (washing and wire brushing) a). The second step consisted of following eight cycles done in the same area reduction value, but with different rolling directions. The roll-bonded initial sandwich was cut into two strips, and after cleaning and finishing treatment, they were stacked over each other. After stacking, they were rotated 90º around the normal direction (ND) of the previous cycle b). The last procedure was repeated eight times at ambient temperature. After the ninth cycles, a multilayered composite with 1792 layers was produced (The microstructures of the specimens produced at odd cycles and their fracture surface after tensile tests were evaluated by scanning electron microscopy (SEM). The SEM examination was done on rolling direction-normal direction (RD–ND) plane. The specimen preparation for metallography included sectioning, mounting, sandpaper grinding, and polishing. The phase analysis were also performed by X-ray diffraction (XRD) using the X′pert high score software. The tests were done in the 2-theta diffraction range of 30–90° with a step size of 0.03° and a step time of 3 s.To investigate the strength and elongation of the fabricated multilayered composites, uniaxial tensile tests were performed according to the ASTME8/E8M-9 standard. The tensile samples were prepared at rolling direction by a wire cut machine. The tensile tests were done at a strain rate of 1 × 10−4 s−1 at ambient conditions using an Instron tensile test machine.The wear test was carried out in accordance with the ASTM G99 standard by the pin-on-disk dry sliding method. Disks with the diameter of 2 cm were prepared with a wire cut machine and used as the wear test samples. Steel pins with the diameter of 6 mm and the hardness of 60 HRC were also used in this study. The pins were placed vertically on the surface of the samples and the wear tests were done at ambient conditions. An accurate digital balance was used to measure the weight of the samples before and after the wear tests.To determine the fracture toughness, compact tension (CT) samples were prepared by using a wire cut machine according to the ASTM E1820–13 standard (shown in ). The fracture toughness tests were carried out by a tensile test machine at ambient conditions. In order to investigate the fracture toughness of the composite samples, three samples were prepared from each cycle and finally their average was reported.Corrosion polarization tests were carried out in a 3.5 wt% NaCl aqueous solution at room temperature. Before the polarization measurements, the prepared specimens were immersed into solution for 40 min to stabilize the open circuit potential (OCP). The reference electrode, counter electrode, and working electrode were Ag/AgCl, Pt, and CARBed specimens, respectively. The values of the corrosion potential, Ecorr, and the corrosion current density, Icorr, were calculated from the intersection of the cathodic and anodic Tafel curves.Changes in the constituent layers of the composites processed by the odd cycles are illustrated in . As can be seen, by increasing the number of CARB cycles, the thickness of the layers is decreased and their number is increased. It is obvious that the layers have changed from smooth to wavy by increasing the strain. Plastic instabilities and irregularities at the interface and in the layers are thoroughly evident from the third pass onwards b and c). The necking in the copper layers occurs earlier than the other two layers because copper has a higher strain hardening exponent c−e), which is consistent with other studies e, the reinforcements (Al and Zn) are relatively distributed in the Cu matrix after final cycles. However, most of Al and Zn layers are continuous, while their thickness is drastically reduced.a depicts the XRD pattern of the multilayered composites produced at one, five, and nine cycles. Also, the Williamson Hall plot of the matrix (Cu) in these cycles is shown in a, only Cu, Zn, and Al peaks are observed in view of the fact that the CARB process was performed at room temperature, and that this temperature is not sufficient to conduct a chemical reaction between/among the layers shows the engineering stress–strain curves of the Cu/Zn/Al multilayered composites processed at the various CARB cycles. As can be seen, the strength increases up to the third cycle and then decreases by increasing the CARB cycles, whereas the elongation decreases continuously from the first to final cycle. The strength in the first cycle is about 311 MPa, in the third cycle it reaches its maximum value of about 330 MPa and at the final cycle it reduces to about 220 MPa. The strength and elongation of the composite in the first cycle of the CARB process is about 1.5 and 7.8 times higher than those in the final cycle. According to the previous reports, the increase of strength during initial CARB cycles is attributed to work-hardening mechanisms and the formation of ultra-fine grains ). The strength of the necked copper layers is not sufficient to withstand the load and resistance to deformation. The formation of small cracks in the interface of the layers can also lead to the decrease in strength. It is but observed that elongation has continuously decreased from the first to the ninth pass. This can be attributed to work hardening, as well as necking and fracture of the copper layers. shows the fracture surface of the primary sheets (Cu, Zn, and Al) after the tensile test and indicates the fracture surface of the Cu/Zn/Al multilayered composites processed at the various CARB cycles. As can be seen, the fracture surface of aluminum and copper consists of dimples, which indicates a ductile fracture, while the absence of dimples in the zinc sheet indicates a brittle fracture. From , it is obvious that the fracture surface of the specimens CARBed at the initial cycles is a mixture of ductile and brittle fracture. The presence of dimples in the aluminum and copper layers in the first and third cycles exhibits a ductile fracture. The characteristic of the ductile fracture is that during the tensile test, the necking, local thinning and small holes in the necked area are first created, then the cavities are connected to each other until they reach a small crack, the crack grows slowly, and rupture finally occurs , the elongation from one cycle to another is reduced, which can also be considered as an emphasis on converting the ductile fracture into the brittle fracture by increasing the applied strain.It should be noted that by increasing strain, the quality of connection between the layers is improved. In relatively higher cycles, the diagnosis of the separate layers and fracture mode is difficult since by increasing the number of cycles, the number of layers increases and the thickness of the layers decreases.a and b demonstrate the samples of the fracture toughness tests before and after testing, respectively. It is clear from b that no separation occurs in the layers of the processed composites during the tests due to the strong bonding.The load vs. displacement curves for the multilayered Cu/Al/Zn composites processed by the odd cycles are presented in . It is obvious from the curves that by increasing the applied strain from the first to the third cycle, Pmax increases, and at higher cycles it decreases severely. As it can be seen, the variation of Pmax is similar to the change of the maximum tensile strength values (see In this study, the R-curve was used to calculate the plane stress fracture toughness in accordance with the ASTM E561 standard. In this method, after performing the plane stress fracture toughness test, the crack length is measured continuously at different forces. This step is the most important part in accurately calculating the amount of plane stress fracture toughness in the method of using the R-curve. For this purpose, according to the standard ASTM E647, the crack growth is measured visually using electron microscopy. In the R-curve method, at a certain crack length, variations in crack growth resistance (Kr) are plotted vs. crack length. Also, the variations in the stress intensity factor (Kapplied) vs. crack length variations at three different levels of constant forces are plotted by a dasher. In this method, the plane stress fracture toughness is calculated from the contact point of the two applied K-curve and the R-curve fiaw=2+aw1−aw320.886+4.64aw−13.32aw2+14.72aw3−5.6aw4In these equations, a is crack size, b and w are the specimen thickness and width, respectively. Also, fi(a/w) is a geometric coefficient in terms of crack growth. These equations are true for a/w≥ 0.35. shows the R-curves of the multilayered composites produced at various cycles of the CARB process. By employing , the variations of the plane stress fracture toughness vs. the number of CARB cycles are presented in . As it can be seen, the fracture toughness of the composite is increased up to the third CARB cycle and then decreased continuously to the ninth CARB cycle. The maximum fracture toughness value belongs to the third cycle (about 31 MPa.m1/2). By comparing , it can be seen that the trend of changes in fracture toughness is similar to that of tensile strength, suggesting that the factors affecting tensile strength also affect the fracture toughness. These observations are in agreement with previous researches , the fracture surfaces of the Al and Cu layers consist of dimples, which represents the ductile fracture of these layers, while in the final cycles dimples are reduced and ductile fracture changes gradually to brittle fracture. Therefore, it is expected that the fracture toughness of the composite samples produced at initial cycles is higher than that of final cycles. Also, it is expected that the fracture toughness decreases gradually from the initial to final cycles.The microstructutal change is one of the parameters which has a negative effect on the fracture tougnhness of the composites fabricated after the third cycle. According to , necking and rupturing occur in the harder layers. The difference in the flow properties of the copper, aluminum, and zinc layers is the main reason for necking and rupturing During the CARB process as a manufacturing method of multilayered composites, the number of layers and consequntly the number of interfaces are increased. The interface between layers can act as places for the initiation of microcracks during the fracture thoughness tests. It has been reported that during the roll bonding process of metallic layers, bonding between the layers does not happen completely and leaves some unbonded areas between them In order to investigate the wear resistance of the composite samples processed at the different CARB cycles, the pin-on-disk tests were performed. shows the variations of the weight loss of the composite samples. As can be seen, by increasing the number of CARB cycles, the lost weight of the samples is increased, and its value in the ninth cycle is 9.5 times higher than that of the first cycle. The detected reduction of wear resistance by increasing the number of CARB cycles is consistent with other studies shows the SEM images of the worn-out surface of the composite specimens. It is observed that the wear trace has become wider and deeper by raising the number of CARB cycles. Two wear mechanisms of abrasion and delamination have been reported for composites processed using ARB and CARB ). By increasing the number of CARB cycles, the number of interfaces between the components of the composites is increased and the thickness of the layers is decreased. Therefore, the nature of the layered structure in the CARB process contributes to the delamination wear type during the wear tests The potentiodynamic polarization curves of the raw sheets and the composite samples processed at the various CARB cycles in the 3.5% sodium chloride solution are shown in . It should be noted that the corrosion tests were performed on rolling direction-transverse direction (RD-TD plane) of the samples. Using the polarization curves, corrosion parameters including the corrosion potential and current density were extracted and listed in a, it can be observed that copper has a more positive corrosion potential and a lower corrosion rate than aluminum and zinc. show that by increasing the applied strain, the corrosion potential from the one to nine cycle becomes more negative. By comparing a and b, it can be seen that the corrosion potential of the primary copper sheet and the composite produced at the first cycle are close to each other. This can be attributed to the fact that the corrosion test has been taken from the surface of the samples, and that in the first cycle the copper is still present on the surface. The macro-images of the surface in the different cycles are illustrated in . As it can be seen, the copper (red region) is only present on the surface of the composite sample produces after the first cycle. By increasing the number of cycles, the thickness of the constituent layers is decreased and the layers of zinc and aluminum move and come toward the surface of the samples (see b, c, and 15d). Therefore, at the higher cycles, aluminum and zinc are gradually present at the outer surface (gray regions) close to copper. The presence of aluminum and zinc next to copper on the surface result in the formation of galvanic cells, decreasing the corrosion resistance of the samples. shows that the corrosion current density of the composite samples increases by increasing the number of CARB cycles. The increase in the corrosion current density indicates a decrease in the corrosion resistance of the samples with increasing the number of CARB cycles , it can be concluded that the composite sample produced at the first cycle has lower corrosion resistance than the primary copper sheet, while the surface layer in both samples is similar (copper). This is attributed to grain refining and residual strain due to the CARB process In this study, Cu/Zn/Al multilayered composites were produced by the CARB process at room temperature. Some properties of the produced composites, particularly fracture toughness, were investigated as a function of CARB cycles and the results are summarized as follows:Al and Zn reinforcements were uniformly distributed in the Cu/Zn/Al multilayered composite after nine CARB cycles.Necking and rupture in the layers were observed after initial CARB cycles. Also, shear bands were created in microstructure during the process.The tensile strength of the composites increased up to the third cycle and then decreased from the third to the ninth cycle. The highest strength (about 330 MPa) and elongation (about 31.5%) values were obtained for the third and first cycles, respectively.The predominant wear mechanism at initial cycles was abrasion, while it was delamination at higher cycles. It was also observed that by increasing the number of CARB cycles, the weight loss increased, as indicative of the decrease in wear resistance.The maximum fracture toughness value was obtained for the third cycle to be about 31 MPa.m1/2.After the third cycle, the fracture toughness of the multilayered composite samples was gradually decreased by increasing the CARB cycles.By increasing the CARB cycles, the Al and Zn layers moved toward the composite surface.By introducing the Al and Zn layers to the surface of the composite samples during the CARB process, the corrosion resistance decreased.Mahsa Avazzadeh: Conceptualization, Methodology, Validation, Investigation, Formal analysis. Morteza Alizadeh: Conceptualization, Validation, Methodology, Validation, Investigation, Writing - review & editing, Project administration, Supervision. Moslem Tayyebi: Methodology, Validation, Writing - original draft.The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.Experimental and theoretical investigation of thickness ratio effect on the formability of tailor welded blankThe influence of thickness ratio (TR) on the formability and forming limit diagram (FLD) of tailor welded blanks (TWB) obtained by pulsed Nd:YAG laser welding of St12 steel sheets are studied. Steel sheets with thicknesses of 0.5, 0.8 and 1.0 mm were combined to form TWBs of different thickness ratios of 2 (1.0/0.5 mm), 1.6 (0.8/0.5 mm) and 1.25 (1.0/0.8 mm). Limit thickness ratio (LTR) is introduced as a new useful factor for prediction of forming limit of TWB. Results of this research show that by increasing the difference of thickness ratio of TWB and LTR, formability and the level of FLD will decrease. In addition, effects of thickness ratio of TWB on the punch's load–displacement, limit dome height (LDH) and weld line movement are studied. By increasing the thickness ratio, the LDH decreases and some defects such as weld line movement and wrinkling increase. The experimental findings of this study show that the thickness ratio of TWB can effect on the position of fracture in TWB products.► A comparative study between different forming limit diagrams (FLD) for tailor welded blanks (TWBs) was done in this research. ► Analytical ductile fracture criterion and some other forming limit criteria were investigated for necking prediction of TWB. ► Load–displacement of numerical criteria was compared with experimental results for different samples. ► Numerical criteria have a good agreement for load–displacement prediction with experiment. ► Advantages and disadvantages of numerical FLDs were presented.Reducing vehicle weight by using plastic, composite and light metallic parts is one of ways to decrease automotive fuel consumption. Weight decrease by using low strength parts is incompatible with parts rigidity and hence automotive safety in accidents will be decreased. Using of TWB in automotive parts is a solution for this incompatibility The limit dome height (LDH) test is used to assess the formability of sheet metal alloys, and is commonly used for steel and aluminum. In this test a dome specimen is produced by a hemispherical punch from uniform sheet. The simplest measure of formability is the height of the dome that can be made before the specimen fractures. Forming height can be increased by some changes in the forming process of TWBs. One solution is increase of thicker (stronger) material flow to decrease the thinner (weaker) material deformation by using different blank holder force. Minor blank holder force is used for thicker materials In this research, effect of thickness ratio of TWBs on the FLD was investigated experimentally. Theoretical part of this research showed that the effect of thickness ratio on the FLD and LDH of TWB can be predicted by LTR parameter. Effect of thickness ratio on the load–displacement of punch and also LDH for different samples is studied too.When the thickness ratio of two materials exceeds a certain value, the material with greater thickness may be limited to elastic deformation or very little plastic deformation in the direction perpendicular to the weld even though the thinner material exceeds its forming limit near the weld. The thickness ratio, in which one material just reaches initial yield strength when the other material reaches its forming limit, is called the limiting thickness ratio (LTR). When the thickness ratio is lesser than the LTR, plastic deformation would occur in both materials. When the thickness ratio is greater than the LTR, the material with greater thickness does not reach its yield point (no plastic deformation) in the direction perpendicular to the weld. This LTR can be determined using a simple mechanics analysis undergo the deformation, the equilibrium equation (without the presence of the friction) in the transverse direction can be expressed aswhere F is force in unit width and subscripts A and B represent the materials A and B, respectively.where σ is the true stress and t is the current material thickness. By using true thickness strain definition and incompressibility law (constant volume law), Eq. where ε1 and ε2 are two true strains parallel and perpendicular to the weld in the sheet plane (). The strain component parallel to the weld must be the same (ε1)A=(ε1)B and Eq. where S is the engineering stress. The above equation provides the relationship of the stress state in materials A and B for the given thickness ratio.Let us assume that material B has greater thickness than material A. It can be shown that SB, reaches the maximum value when SA, reaches the maximum value. The maximum value for SA is the tensile strength of material A. Therefore, the LTR can be obtained when the flow stress SB in material B reaches the yield strength of the material and SA reaches the tensile strength of material A. Eq. where STA, is the tensile strength for material A and SYB is the yield strength for material B. Hence, the LTR can be obtained asThe LTR depends upon the tensile strength in the lesser thickness material and the yield strength in the greater thickness material The experimental materials used in this study are St12 steel sheets. The chemical composition of the steel grade used in this work is given in Mechanical properties of St12 steel sheet was characterized by standard uniaxial tensile test of ASTM-E8 at 2 mm/min cross-head speed shows comparison of stress–strain curves for different thicknesses of St12 sheets that were used in this study. shows the mechanical properties and forming parameters of the mentioned steel sheets. Hollomon's equation (σ=Kεn) was used to model the plastic behavior of the sheet materials. For calculation of n and K, statistical analysis was used to assess the best order of polynomial. Last column of shows the R2 values for curvature fitting of stress–strain curves respected to steel sheets with different thicknesses. For all three thicknesses of the St12 sheets, it is found that R2>0.95, which indicate how well curves fit the data.For preparation of TWB with different thickness ratio from St12 steel, pulsed Nd:YAG laser welding was used. In this welding process no filler material was used. Three different thickness of 0.5, 0.8 and 1 mm of St12 sheet were used to weld together and make TWB with thickness ratios of 1.25, 1.6 and 2. The thickness combinations and the different thickness ratios of TWBs are shown in . Model IQL-10, a pulsed Nd:YAG laser with a maximum mean laser power of 400W was used as welding laser source for the experiments. shows the laser welding setup. Square shape pulses are the standard output of this laser. Pure argon gas was blown coaxially to protect the welding region. Details about the laser welding setup used in this study can be found elsewhere Besides forming parameters and thickness ratio of TWB which affect the forming behavior of TWB, welding parameters are very important to produce a qualified TWB part. The pulsed laser beam welding process is controlled by a variety of parameters. In the other words, welding parameters should be selected such that fracture does not happen from the weld region. These parameters for Nd:YAG laser welding are: average power, laser's head distance with sheet surface, laser frequency, welding speed and pulse duration.In a part of this research, Taguchi approach shows the experimental parameters and their levels for Taguchi design of experiment method. For example, in the 3rd row there are three levels of laser speeds, 4, 5 and 5.84 mm/s. Two formability standard tests of uniaxial tensile test (ASTM-E8) shows standard ball punch test setup which used for evaluation of sheet formability. The objective of standard ball punch formability test is to determine the forming height, which is a function of the process parameters. Substandard transverse welded tensile specimens were cut from the laser welded samples for uniaxial test of the TWBs. shows parameters arrangement for these 18 welding tests. Last column of shows the forming height of ASTM-E643 test. Forming height has been the maim criterion for selection of laser welding parameters.The biaxial stretch forming tests were done according to the procedure suggested by Hecker Seven specimens of size 25×175 mm to 175×175 mm for FLD of every thickness ratio were cut from the laser welded specimen such that weld line were perpendicular to the stretching direction (transverse specimens). FLD tests were repeated three times for each thickness ratio of TWB, so 21 samples were used for every thickness ratio. Specimens were grid marked with circles of 2.5 mm by electrochemical etching method to measure major and minor strains after deformation. The circles on the sheet samples became ellipses after deformation, falling into safe, necked, and failed zones. The principal strains in the plane of the sheet were expressed using the true strain measures. The true major and minor strains were calculated by measuring change of the principal directions, a and b, of the ellipse with reference to the initial diameter, d0, of the circle:where a and b are the ellipse diameters and d0 is the initial diameter of circle. FLD was drawn by separating the safe limiting strains from the unsafe zone containing the necked and fractured ellipses.As mentioned before, Taguchi design of experiment and two formability tests of ASTM-E8 and standard ball punch were used for choosing correct parameters of laser welding. ASTM-E8 is an in-plane uniaxial tensile test, but standard ball punch test is an out-of-plane formability test which considers the effects of tension and bending. Incorrect selection of welding parameters can influence on the position of fracture of TWB. shows two transverse TWB samples after uniaxial tension test (ASTM-E8). shows results of standard ball punch test (ASTM-E643) on the TWB samples. Fracture happened in the thinner side of TWB and far from weld line for the samples which welding parameters were selected correctly ((a)), but for incorrect welding parameters fracture happened in the weld line ( shows proper laser welding parameters which were obtained based on the results of design of experiment and used in this study for welding of TWB samples.The load–displacement curves of the TWBs obtained through the data acquisition system during stretch forming in experimental tests. shows the effect of thickness ratio on the punch's load–displacement of TWB with different width. shows the LDH values of TWBs with different thickness ratios and different width. obviously indicate when the thickness ratio of TWBs increase, LDH of samples decrease and this trend is repeated for all samples with different width.LTR, thickness ratio (TR) and their difference (Δ=TR−LTR) can be calculated by analytical result (Eq. LTRTWB(0.5−1)=(ST)0.5(SY)1=325.25189.2=1.72,TR=10.5=2,Δ=TR−LTR=0.28LTRTWB(0.5−0.8)=(ST)0.5(SY)0.8=325.25191.92=1.69,TR=0.80.5=1.6,Δ=TR−LTR=−0.09LTRTWB(0.8−1)=(ST)0.8(SY)1=324.44189.2=1.715,TR=10.8=1.25,Δ=TR−LTR=−0.465, thickness ratio in the case of TWB with TR=2 exceeds the LTR (Δ=0.28), for TWB with TR=1.6 this difference is Δ=−0.09 (TR<LTR) and for TWB with TR=1.25 this difference is Δ=−0.465(TR≪LTR). Experimental LDH of TWB with different thickness ratio and these analytical results comparison show that by increasing of Δ, LDH decrease. shows the variations of LDH vs. Δ for TWBs with different widths. indicates that decreasing of LDH with increasing of Δ happens for all TWB samples with different width. shows that TWB specimens with different widths yield different LDHs. This trend is the same for all TWB samples with different thickness ratio. For all three TWB with different thickness ratio least of dome height is occurred for 150 mm width sample. This figure showed that minimum of LDH was happened near the plan strain condition. This result is agreed with the result of Hosford shows the FLD of TWBs with different thickness ratios. It indicates that TWBs of the highest thickness ratio (2) yield the lowest FLD level, while TWBs of the lowest thickness ratio (1.25) yield the highest FLD level. This figure shows that by thickness ratio increasing, level of FLD decrease. shows the variations of minimum major strains (FLD0) of TWBs vs. the parameter of Δ. It demonstrates the trend that a higher values of Δ a lower value of the minimum major strain in the FLD. This also implies that the higher the Δ, the lower the formability of TWB. shows the fracture position of the TWBs with different width for thickness ratio of 2 (TWB 1−0.5). The fracture of all TWBs samples was parallel to the weld line and perpendicular to the principal strain. It occurred at the thinner part of the TWBs. Generally, a thinner material resists a much smaller force than a thicker material. When the thinner material undergoes plastic deformation, the thicker one is still deforming in the elastic region.Fracture parallel to the weld is caused by too large thickness ratio of base blanks of TWB and the splitting occurs in the side of material with lesser thickness. For this type of fracture, it can be predicted that the plastic strain in the direction perpendicular to the weld, at the thicker material side of the weld, would be small or zero for the welded blank with large thickness ratio.If the materials of TWB had the similar or less difference thickness (or when the Δ decreases), the plastic deformation which now is confined to the lesser thickness material alone would normally be shared by the materials on both sides. When the thickness ratio or Δ is decreased, the position of fracture from parallel and near to weld line is changed to perpendicular and far from the weld line. shows the position of fracture for TWB samples with different thickness ratio and also an unwelded blank. This figure indicates that the position of fracture for TWB with minimum thickness ratio is similar to unwelded blank.Wrinkling of TWB parts is another defect which thickness ratio of TWB influence on it. obviously indicates that by thickness ratio increasing, the number of winkle in the edge of TWB samples increase.Movement of weld line during deformation of TWBs is important and can influence on the forming behavior of TWB sheets. When the load is applied, the thinner side resists lesser and hence deforms to a more extent compared to the thicker side. This isolated deformation results in significant increase in the strain level in the material with lesser thickness and the maximum strain would occur in the area just adjacent to the weld. Due to this the weld shifts towards the thicker side, allowing more strain on thinner side compared to the thicker side. obviously show movement of weld line which is towards the thicker side of TWB. shows the variation of weld line movement for TWB with respect to distance from the edge for samples with width of 175 mm. As this figure shows weld line movement increased by thickness ratio increasing. Maximum of weld line movement is at the pole of TWB samples.As a result, the weld line would move in the direction of the material with greater thickness because of the deformation on one side of the weld only. This isolated deformation results in significant increase in the strain level in the material with lesser thickness or lower strength and the maximum strain would occur in the area just adjacent to the weld. When the maximum strain exceeds the sheet's forming limit, splitting occurs adjacent to the weld in the thinner material.In this study, the mechanical properties of the TWBs obtained by pulsed Nd:YAG laser welding of St12 sheets with different thickness ratios were investigated. The experimental FLD of TWBs were determined based on the Hecker test set-up for different thickness ratios and compared. Biaxial stretch forming of Hecker was used for the FLD investigation of TWBs. Theoretical analysis was used to predict the formability of TWB and the effect of thickness ratio on the FLD of TWB. Following conclusions can be drawn from this study:Parameters of Nd:YAG laser welding have significant effect on the weld quality of TWBs. By choosing proper welding parameters, weld quality of the TWB in the forming process can be controlled.Thickness ratio of TWB is an important parameter which influence on the FLD, fracture position and wrinkling of TWB parts. Limit thickness ratio (LTR) is a useful and practical ratio for TWB's forming limit specification. When the difference of thickness ratio and LTR increase, LDH and minimum major strain (FLD0) decrease. Results of this research pointed out that the higher thickness ratio, results to the lower FLD levels. In addition the results show that the higher the thickness ratio, the lower is the punch's load–displacement.Thickness ratio can effect on the position of the fracture. Fracture parallel to the weld is caused by too large thickness ratio of base blanks of the TWB and the fracture occurs in the side of thinner material. When the thickness ratio decreases, position of the fracture will move far from parallel and near to the weld line.Wrinkling and weld line movement are other defects in the forming of TWBs. Number of wrinkles and also weld line movement are increased by increasing the thickness ratio. Maximum movement of the weld line is at the pole of TWB samples.Interlayer reinforcement of 3D printed concrete by the in-process deposition of U-nails3D concrete printing has tremendous potential for construction manufacturing; however, weak interface bonding between adjacent layers remains a well-known issue that affects the mechanical properties of printed structures. The layers introduce anisotropy and reduce the capacity to resist tensile and shear loads. Reinforcements, inserted perpendicular to the printed layers to traverse the interfaces, can improve these limitations, but the insertion of reinforcements is difficult to achieve in practice, and there are few published studies exploring appropriate methods. This study presents a promising approach using U-shaped nails inserted into concrete during the printing process. The bridging effect and dowel action of the applied U-nails are visualised and analysed to elucidate the toughness improvement. The ultimate tensile strength and shear strength of 3D printed concrete are significantly increased by 145.0% and 220.0%, respectively. U-nails with a filament thickness of 2–2.5 mm are recommended to yield optimal improvement in the interlayer strength.In recent years, 3D concrete printing (3DCP) has become an important technology in digital fabrication with concrete (DFC) methods []. 3DCP has an advantage in creating bespoke components because it eliminates the need for a mould, and thus it is cost-effective for low-volume manufacturing. It has become a steering index of digital construction with numerous successful applications in practical engineering, such as 3D printed bridges, bus stations, public toilets, and villas, reflecting the strong development potential of 3DCP technology []. In October 2019, a 3D printed concrete bridge with a single span of 18.04 m was built on the Beichen Campus of Hebei University of Technology, Tianjin, China as shown in . This structure is credited with a number of innovations in 3DCP for load-bearing structures.However, 3DCP is still in its nascent stages, and many issues related to the materials and structural properties are yet to be fully understood []. In particular, the increasing applications of 3D printed concrete as stressed structures have highlighted the poor tensile strength and toughness of concrete. Thus, the use of reinforcements is mandatory for most structural applications. This necessitates the reinforcement of printed concrete for structural applications []. For 3DCP technologies, owing to the distinct layered stacking process, traditional reinforcement methodologies cannot be directly implemented. Underperformance in reinforcing remains a serious constraint for the promotion and application of 3DCP.The layer stacking process of 3D printing is likely to incur weak interfaces owing to different curing rates and evaporation of surface moisture. This phenomenon negates the uniformity and integrity of printed materials, and thus adversely affects the mechanical strength, toughness, and durability of the printed structures []. Significant effort has been dedicated to the strengthening and toughening of 3D printed concrete by researchers, as elaborated below.A prevailing method for strengthening 3D printed concrete materials is the introduction of short high-modulus fibres to provide strain-hardening performance []. To reconcile with the 3D continuous extrusion procedure, short fibres with considerable rigidity can be introduced to foster an orientational effect aligned with the printing direction for nozzles with small print heads. This process provides full exertion of the constraint of the fibres on the material deformation to improve its strength and toughness []. However, owing to the discontinuity of the short fibres, their functionality is limited to improving the shrinkage and crack resistance performance.Some researchers have explored the direct deposition of reinforcements concurrently with the 3D printing procedure. Francesco et al. [] presented experimental results on the flexural reinforcement of a geopolymer mortar with additively manufactured metallic rebar using electron beam melting. Mechtcherine et al. [] 3D printed steel reinforcements for digital concrete construction using gas–metal arc welding. The 3D-printed rebar displayed better ductility than conventional B500 steel bar. Xu et al. [] proposed an alternative approach for creating strain-hardening cementitious composites (SHCC) by applying 3D printed polymeric reinforcement meshes. Combining 3D printed reinforcements and concrete composites is an effective way to improve the automation and intelligence of building construction. However, there is still a lack of research on the long-term service performance and anti-aging properties of 3D printed metal and polymer materials. In particular, the cost remains relatively high.For the reinforcement method in the vertical direction, ETH [] used robots to automatically fabricate a dense welded steel mesh on site, and the transverse steel bars were designed in a discontinuous form to adapt to the model curves. Special concrete materials were poured into the steel mesh to prevent the concrete from flowing out of the mesh grids and to realise self-compaction. At the same time, shotcrete was used for external shaping and surface smoothing. Jay [] presented a mesh reinforcement method for 3D concrete printing by designing a nozzle with a fixed height to deposit the reinforcing mesh vertically. Good mechanical properties were demonstrated through three-point bending tests. Jay Sanjayan [] also designed a layer penetration reinforcement method to provide interlayer reinforcement. An increase of 184% in the flexural strength of the printed beam elements was measured after penetration of deformed bars. Omar Geneidy et al. [] proposed a promising strategy for in-situ printing that vertically embedded “staple” reinforcements to provide interlayer interlocking. An increase in the tensile properties was demonstrated in three-point bending tests. Jacques Kruger et al. [] presented a method of manually inserting 50-mm-long straight steel fibres into the concrete during printing to facilitate ductile failure and increase the interlayer mechanical properties. The post-peak mechanical ductility was observed to yield under deflection softening. Niklas Freund et al. [] proposed an automated and robot-guided integration of short reinforcement bars perpendicular to the layer direction. Both the surface roughness of the reinforcement bars and the early age stiffness of the printed concrete affect the bonding quality between the reinforcement and matrix, and an improper direct insertion process may introduce voids in the interlayer bonding surface. Lauri Hass and Freek Bos [] presented a novel reinforcement method that introduced a linear reinforcement element with a helix-shaped surface accentuation into fresh concrete. Desirable bonds and few defects between the reinforcement and matrix material were confirmed through pull-out tests.For a horizontal reinforcement method, Winsun [] manually incorporated a steel mesh into the interlayers for practical engineering applications. To adapt to the twists and turns of the printing path, a special device can be designed and installed on the printing nozzle that synchronously follows the movement of the printing nozzle to deposit reinforcing materials such as steel bars into the concrete. Thus, the reinforcement materials and concrete can be integrated into the composite as a whole. The continuous and simultaneous entrainment of flexible reinforcements into the 3D printing process should be explored to satisfy the mechanical property requirements. Lim et al. [] presented a hybrid reinforcement approach using both short fibres and in-process steel cables for reinforcement, and a 290% increase in the flexural strength of 3D-printed concrete was reported. Freek P. Bos [] proposed a hybrid down/back-flow nozzle to optimise the cable–concrete bond properties, and the corresponding design was successfully applied to print a full-scale bicycle bridge. We have also previously proposed a promising approach for simultaneously entraining a continuous micro-steel cable (1.2 mm) during a filament (12 mm) deposition process to fabricate reinforced composites. The integration and optimisation of the printing process for geopolymer materials with the cable-inlaying process have been explored and validated []. This method can fully adapt to the flexible advantages of 3D printed concrete and provide horizontal reinforcement. However, it failed to improve the weak surface between layers.One post-installation reinforcement method employs 3D printed concrete formworks based on contour crafting technology. After material curing, the reinforcement is manually deployed, and fresh concrete is then poured to fabricate a complete structural component. Asprone et al. [] attempted to assemble a series of 3D printed hollow components with the engagement of external reinforcement anchorages to construct two beams of approximately 3.0 m in length. Lim et al. [] used post-tensioned pre-stressing tendons to toughen 3D printed concrete by reserving holes for the prestressing tendons in the printed members to improve the mechanical strength and toughness. This method is simple in application and provides effective reinforcement, but it is difficult to adapt to the non-straight printing paths of 3D printed structures for rigid reinforcement. Manual operation is required, thus negating the degree of automation.Based on the above review and analyses, each of the currently available reinforcement methods has unique advantages. However, most of the reinforcement methods do not synchronise with the 3D printing process for concrete materials; thus, they are limited in automation and flexibility. In particular, there is little data available on synchronous reinforcement methods for the weak planes of layered printing structures.Taking advantage of the readily available reinforcement equipment in our laboratory, micro-cables can be entrained in the material deposition process parallel to the designed printing path. A vertical reinforcement approach concurrent with the 3D printing process is necessary to improve the interlayer bonding and produce a combined reinforcement effect to complement our previously proposed micro-cable reinforcement method. U-nails are designed and vertically deposited to provide interlayer reinforcement and avoid collisions with the embedded micro cables. The U-nails are installed perpendicular to the weak plane of the printed layer to increase the interlayer bond strength of 3D printed concrete. Therefore, this study is committed to providing a solution, including both software and hardware, for designing and developing a synchronous reinforcement system that can be implemented simultaneously with the printing process. Further, this study aims to promote the general practical application of 3D concrete printing technologies. shows an image of our 3D printing system. It consists of a print head, concrete transmission pipe, peristaltic pump, robotic controller, and printing platform. An industrial Kawasaki 6-axis robotic arm is used to facilitate flexible spatial movement; it features a horizontal working space of 1.73 m, positional accuracy of ±0.05 mm, maximum linear moving speed of 11.5 m/s, and load capacity of 20 kg. All of the other printing hardware is self-designed to cater to specific printing demands. Before the beginning of the printing process, the raw materials are first added to the premix mixer, which directly deposits the mixture in the chamber of the concrete pump after the blending process is complete. The printing process starts when the prepared mixed mixture is transported through the concrete pipe to the print nozzle. The pump, robotic controller, and print heads are cooperatively controlled in the printing process. The diameter of the nozzle is 10 mm.To facilitate the automatic insertion of U-nails, a mechanical device is designed to be fixed on the print head. As illustrated in , the U-nail insertion device is assembled at the material container of the print head, which utilises a square printing nozzle to smooth the outer surface of the extruded filaments, thereby ensuring the printing quality. In particular, a 90° elbow-shaped nozzle is designed to facilitate the insertion of U-nails concurrently with concrete extrusion by shortening the distance between the nozzle and the position of the inserted U-nails. The material feed inlet is located at the top of the material container. Servomotor A is used to drive the blade inside the material container for concrete deposition, while the premixed concrete enters the material container from the feed port. Servomotor B controls the rotation of the printing nozzle using a gear transmission system. The rotation of the nozzle steers the U-nail insertion to conform to the movement of the print head following curved printing paths. Therefore, the automatic nailing function can adapt to the printing path. The rotation angle and speed of the nozzle are controlled by Servomotor B and the gear transmission system, which are independent of the upper material extrusion system.(b), the key machinery for U-nail insertion consists of an electromagnet exciter, firing-pin head, U-nail storage box, slider, and spring. The firing-pin head, which is connected to the electromagnet exciter, can quickly move down and drive the U-nails into the concrete through the magnetic force generated by the electromagnet exciter. Thus, the U-nails are inserted vertically in the fresh mixture. A time relay is installed on the switch of the electromagnet excitation device to trigger the release of U-nails at specified time intervals. The cross-section of the firing-pin head matches the top surface of the U-nails to ensure appropriate contact. The frequency of U-nail insertion is controlled by the operation of the electromagnet exciter. During printing, the next U-nail in line is pushed forward to the outlet of the nail storage box by the slider and the spring after the previous U-nail has been inserted into the concrete.Most part of the schematic design have been realized in this section except the elbow shaped square printing nozzle, where a circular shaped nozzle illustrated in shows photographs of the assembled and disassembled U-nail device. The device is composed of the shooting system (iron core, spring, electromagnet, U-nail storage) and the control system (master switch, time relay, power control knob), both of which are connected and controlled by the control chip. The power control knob controls the magnetic force of the electromagnet by changing the magnitude of the input current to control the insertion force of the U-nails. The knob is connected to the control chip. The master switch ensures the safety of the device and controls the on-off state of the entire system. The system is an accessory device for the 3D printing extrusion system; as the U-nail device is not integrated with the path planning software of the 3D printer, manual control of the power control knob and master switch is required. Before the experiments, the distance between the device and concrete surface and the penetration force controlled by the power control knob need manual adjustment through trial-and-error to determine the acceptable settings. In the printing process, the insertion frequency can be controlled by the time relay, while the on-off state of the system still requires manual operation.This section provides a workable and detailed solution design of cooperative control between U-nail insertion and 3D concrete printing. The extrusion of concrete and movement of the robotic arm are controlled by the path information file, which is generated by inputting the printing process parameters in the sliced 3D digital model. All of the point distribution information in the printing path, printing and extrusion speeds, and starting and stopping settings are stored in the path file, which is read by the operation system of the 3D printer to govern the movement of the robotic arm. Conventionally, the path file contains only the process information for the concrete and printer. With the introduction of the U-nail insertion function in the printing process, cooperative control is required to reconcile the settings of the U-nail insertion into the path file. The parameters for U-nail insertion to be specified include the horizontal deposition speed/frequency, vertical insertion depth, and rotation scheme of the 90° nozzle. The working diagram of the U-nail insertion system during the printing process is shown in . It should be noted that the rotation axis of the system is set at the outlet of the square print head.Horizontal deposition spacing: The horizontal deposition spacing of the U-nails, lh, is directly related to the insertion frequency or insertion time interval, Δt, under a fixed printing speed, vp. The spacing between successive U-nails can be adjusted by setting different time intervals. The relationship can be expressed as follows:Vertical insertion depth: The insertion depth of the reinforcements, ld, is equal to the leg length of the U-nails, lnail. It should be noted that the U-nails cannot be inserted in the first deposited layer because the length of the U-nail is greater than the thickness of each layer, Δh, to allow the nails to bridge the interface between two adjacent layers. The parameter to be determined is the layer in which the U-nails are first to be inserted; the number, N, of layers for U-nail insertion and the corresponding structure height, H, are calculated as follows:The setting of the layer number, N, or the height of the structure, H, for the start of U-nail insertion in the path file should be greater than the calculated values.Rotation of the nozzle. Limited by the 90° elbow shape, the nozzle rotates to adapt to the curved printing path, which is specified by a series of points and corresponding position information with (x, y, z) coordinates. The movement direction of the print head is determined by the vector of two adjacent points on the printing path, as shown in (b). For example, the print head is designed to move to Point C (xc, yc, zc) from Point A (xa, ya, za). The travel time (tAC) is determined by the curved distance, which is calculated as the sum of distances |AB| and |BC| and the pre-set linear printing speed, vp, as follows:The rotation angle of the nozzle can be calculated using the coordinates of three related points as follows:The angular velocity (ωB) of the print head passing through the three points A–B–C is then calculated as follows:Rotation of the square nozzle is controlled by servomotor B. shows a flow chart of the 3D concrete printing procedure with concurrent U-nail insertion. It can be divided into main three stages. First, the 3D digital model (STL format) is sliced, and the printing path and printing process parameters are specified; the pre-G-file with point coordinates and relevant information is then output. In the second stage, the deposition parameters of the U-nails are specified to coordinate with the concrete printing, and the operation file that includes the printing process parameters, material transport parameters, and U-nail layout parameters is output. Finally, the 3D printer reads the operation file to begin printing. The code of this solution, i.e. Eqs. has been finished in the path planning software, while the control link between the digital software and the 3D printer hardware has not been realized. The transition from software to 3D printer hardware still need further works of automation professionals. Then in the actual production processes of the specimens, manual control of the power control knob and master switch have to be employed to serve as the control link.A cement-based composite material catering to extrusion 3D printing is prepared according to our previous study []. Specifically, ordinary Portland cement (OPC) and high-belite sulphoaluminate cement (HB-CSA) are mixed. The water–cement ratio is set as 0.4, and the cement–sand ratio is 1.5. The initial setting time of the mixture is adjusted to 60–80 min, and appropriate amounts of superplasticizer and silica fume are added to ensure the consistency and extrudability of the mixture. The mix proportions are listed in . Dry powder materials are first added to the mixer according to the designed proportion and mixed for 10 min. The superplasticizer agent and water are then separately mixed and poured into the dry mixture before mixing for 6 min, after which the mixture is ready for printing.To assess the reinforcement effect of the U-nails on the interface between layers, specimens with dimensions of 100 mm (L) × 100 mm (W) × 100 mm (H) are fabricated for split tensile tests, and specimens with dimensions of 200 mm (L) × 100 mm (W) × 100 mm (H) are fabricated for double shear tests. A contour offset path is specified for printing the samples. The nozzle has an opening of 10 mm. The linear printing speed is set as 45 mm/s, the extrusion speed is 2.7 cm3/s, and the printing thickness is 6 mm. The layout of the U-nails in the 3D printed samples for the experiments is shown in . The dimensions and geometry of a single U-nail are illustrated in . The width, length, and thickness of the nails are 6 mm, 22 mm, and 1 mm, respectively. U-nails with thicknesses of 2 mm and 3 mm are created by connecting two and three single U-nails together, respectively. The surface of each U-nail is smooth.In the mechanical tests, different numbers of U-nails are bundled together to create different nail thicknesses for a single insertion to enable varying reinforcement effects. In this study, the variation in the reinforcement effect of U-nails on the interlayer tensile and shear bonding properties is evaluated through a three-level factorial design by defining different distribution spacing and U-nail thicknesses, as summarised in Three spacing intervals (l0) (20 mm, 30 mm, and 40 mm) and three thicknesses (φ) of U-nails (3 mm, 2 mm, and 1 mm) are combined in a matrix to constitute nine different reinforcement levels. Meanwhile, a group of unreinforced printed samples is prepared to serve as a control, denoted as CC. The length of each nail is 22 mm, which is between three and four times the printing layer thickness. Therefore, the first layer into which U-nails are inserted is the fourth layer. To enhance the reinforcement effect, U-nails are inserted in every second layer with a staggered layout, as shown in ; a typical U-nail reinforced 3D printed sample is shown in . In the sample manufacturing process, the linear printing speed is 45 mm/s, and the time interval for each printing layer is thus calculated as 40 s based on the entire printing path for each layer. The printed samples are cured at 20 ± 2 °C and 95% relative humidity for 28 d.The U-nails are designed and incorporated to improve the bonding between interlayers. The damage that occurs at the interfaces mainly comprises tensile and shear cracks due to separation or slippage. Therefore, tensile and shearing tests are conducted on the interfaces to evaluate the improvement in bonding provided by the U-nail reinforcement. The tensile and shear stresses are assumed to be constant over the entire surface for both tests.The splitting tensile test method is applied owing to its easy operation process. The setup for the tensile tests is shown in . The samples for testing are cut from the sample shown in , and the cutting surfaces are smoothed to ensure accurate loading. Square cushion strips (5 mm × 5 mm) are placed on the top and bottom of the samples parallel to the printing layer direction. The load is continuously increased at a loading speed of 0.06 MPa/s until the printed sample undergoes splitting failure. The splitting tensile strength (ft) is calculated as follows:where Ft is the peak load (N), and A is the load-bearing area (mm2).], double-shearing tests were used to measure the shear strength of 3D printed concrete. This method is easy to perform without requiring extra effort to prepare irregular samples or a complex setup. The ratio of the cross-sectional height to specimen length is designed as 1:2 for shear testing. Specimens with dimensions of 200 mm (L) × 100 mm (W) × 100 mm (H) are fabricated for the double-shear tests. Similar to the splitting test, the printing layers are aligned parallel to the shear loading direction. As shown in (b), steel plates with similar thicknesses are aligned and placed on the top and bottom of the test sample to ensure that the shearing force passes through the interfaces of the printing layers. The loading speed is set to 0.1 mm/min to obtain the force–displacement curve, and the shear strength (fτ) is calculated as follows:where Fτ is the peak shear load (N), and A denotes the shearing area (mm2). It should be noted that the area (A) of the irregular cross-section formed as a result of printing imperfections should be converted to the equivalent rectangle for calculation. shows the results of the interface tensile and shear strengths. The results show that the interface bonding performance of the U-nail reinforced samples is generally better than that of the unreinforced counterpart. As shown in (a), the interface tensile strength is positively correlated with the thickness of the U-nails and negatively correlated with the spacing interval. The interface tensile strength and peak strain are reduced by increasing the distribution spacing for identical U-nail thicknesses. When the nail spacing increases from 20 mm to 40 mm, the corresponding tensile strengths for U-nails with thicknesses of 1 mm, 2 mm, and 3 mm decrease by 42.8%, 49.3%, and 38.9%, respectively. For a similar layout spacing, the interface tensile strength increases by 26.3%–35.3% when the U-nail thickness is increased from 1 mm to 3 mm. Additionally, based on the testing results, the splitting tensile strength of unreinforced cast concrete is 2.12 MPa, which is 5.2% higher than that of the printed concrete specimen (CC). The nail reinforcement can improve the performance of printed concrete with short time intervals (40 s in this study). Its effectiveness for longer time gaps still requires further testing.(b) show that the shear strength and peak strain of the printed sample reinforced by nails of similar thickness decreases with an increase in the spacing interval. According to the relationship in Eq. , the shear strain (γ) of the nails is controlled by the sustained shear stress (τs) and inherent shear stiffness (G). For identical thicknesses, the nail stiffness is the same. The larger the layout spacing of the nails, the higher the sustained shear stresses will be. The induced shear stain of the nails increases with the spacing, and the toughness effect is therefore weakened. For U-shaped nails with thicknesses of 1 mm, 2 mm, and 3 mm, the shear strength decreases by 34.6%, 22.1%, and 24.5%, respectively, when the spacing interval increases from 20 mm to 40 mm, as shown in (b). For an identical spacing, the sustained shear stress of each nail is the same. The larger the thickness of the nails, the higher the shear stiffness will be. The induced shear strain of the nails decreases with the thickness; therefore, the toughness effect is improved. However, for a similar spacing layout, U-nails with a thickness of 2 mm exhibit the best shearing reinforcement effect on the interlayer bonding, rather than the reinforcement effect increasing with the thickness of the U-nails. This is probably because nails with a small thickness are prone to deform along with the cement matrix when exposed to extra loading. However, when the stiffness of the nails reaches a certain high level, the cement matrix fails first without inducing significant strain in the nails, and the shearing capacity decreases correspondingly. depicts the interface splitting tensile stress–displacement curves for the U-nail reinforced concrete specimens. It is understood that the applied splitting tensile stresses are resisted by the coupled actions of interfacial contact bonding and nail–concrete cohesion. In the test results, the unreinforced specimen (CC) displays obvious brittle failure. Once the peak load is reached, the stress decreases sharply. On the other hand, for the nail-reinforced specimens, the tensile capacity increases with increasing nail thickness and decreasing spacing intervals, exhibiting distinct tensile resistance. In (a), U-nails with a thickness of 1 mm produce minimal improvement on the interlayer tensile strength, and specimens 1N20 and 1N30 show comparable strengths to that of the non-reinforced specimen (CC). The enhancement becomes notable when the deposition interval is reduced to 20 mm. The interlayer tensile strengths of specimens 2N20 and 3N20 are 37.8% and 61.8% higher than that of specimen CC, respectively. The displacement at the peak load differs significantly between samples and the regularity of the mechanical characteristics is not clear. This may be due to the manual operation of the nail penetration in these experiments, causing human factors to have a certain impact on the results. represent the pull-out process of U-nails from the cement matrix, consistent with the experimental observations. shows the interface shear tensile stress–displacement curves for the U-nail reinforced concrete specimens. In contrast to the tensile resistance behaviour, the shear resistance of the U-nail reinforced samples is generally better than that of the unreinforced sample (CC). The embedded nails resist the applied shear stresses through the bridging of adjacent layers and dowel action. Thus, the resistance of nail-reinforced composites against shear deformation is improved. In this study, dowel action refers to the forces transferred perpendicular to their axes. The shearing deformation and interfacial slippage can be counteracted, and thus restricted, by the perpendicular U-nails.Before the peak shear load is reached, the concrete and U-nails jointly sustain the applied loads, and the deformation of the concrete is restricted by the embedded nails. The pre-peak curves of the samples all exhibit a similar increasing trend. The crack resistance ability is improved by the U-nails, as indicated by the post-peak curves. In the post-peak curves in (b), the shear loads initially decrease rapidly and then decrease more gradually. In the uncracked stage, the applied shear loading is resisted almost entirely by the concrete. Once the shear strength of the concrete is exceeded, shear cracks occur, and the load rapidly decreases owing to the inherent brittleness of concrete. Then, the shear stress is transferred to the perpendicular nails, which may deflect or distort to dissipate the shear deformation. Cracks are intercepted, and therefore the decrease of the loads is smooth.The area under the load–deflection curve, particularly in the post-peak stage, can be used to estimate the energy dissipated during the crack propagation process, which can also be considered as an index of the toughness. Qualitatively, the area under the post-peak stage increases with the nail thickness for identical spacing intervals. However, this method is not yet available for the shear toughness analysis of U-nails, which requires further study and evaluation. shows the mesoscopic structures of the U-nail reinforced specimens after splitting failure. The central load-bearing zone is extracted for computed tomography (CT) scanning ((a)). Slices from two orthogonal directions are presented to observe the effect of the U-nails on adjacent interlayers. (b) shows two arbitrary slices from the YZ plane perpendicular to the splitting surface. The macro-cracks induced by the splitting stresses are clearly displayed. The splitting stress acts parallel to the Y direction, and the cementitious matrix separates in the Z direction. As illustrated in (b), the nails (light grey colour) are observed to bridge between both sides of the crack, enabling them to resist the widening of cracks and separation of the matrix. Therefore, a de-bonding phenomenon between the nail and cementitious matrix occurs to dissipate the splitting energy through interfacial friction forces. As shown in (c), slices are extracted from the XZ plane, which is also perpendicular to the splitting surface. The U-nails are distributed across the splitting cracks, and the process of U-nails pulling out from the cementitious matrix can be distinctly observed.In particular, meso-cracks are observed at the connecting end of the U-nails, which are parallel to the macro-cracks. In contrast, no cracks are observed at the free end of the U-nails. The meso-cracks are likely caused by the tensile stresses transferred to the nails through the splitting action of the concrete sample. Tensile stresses drive the pull-out of the nails. The connecting end of the U-nails favourably increases the bonding between the U-nail and matrix, producing an anchor effect. The U-shaped nail produces a similar effect of restraining deformation and crack development as a straight nail, because the roots on each side of the crack are susceptible to pulling out in the same manner as a straight nail. shows the failure pattern of the prism samples exposed to shear loading; straight shear cracks are observed, demonstrating the insignificant influence of flexural components on the shear tests. (b–c) shows CT slices perpendicular to the shearing plane. The U-nails are distributed across the shear cracks, which are generally smaller than those formed after the splitting failure. In the scanned results, some nails have de-bonded or separated from the matrix to dissipate the transferred shear stresses. From this perspective, the incorporated U-nails lack sufficient interfacial bonding with the cementitious matrix. On the other hand, the embedded U-nails produce distortion or staggered deformation under shear loading and toughen the cement matrix to resist shear deformation. In contrast to the splitting failure mode, no pull-out of the nails is observed from the free end, and no obvious minor cracks are found near the connecting end of the U-nails. Therefore, the U-nails embedded at the interfaces of the deposited layers perform a distinct dowel reinforcement function in resisting the shear loading. The friction forces between the nail and cement matrix have an insignificant contribution to the shear reinforcement of the nails to the interlayer bonding. The dowel function of the U-nails depends on the inherent rigid properties, which is different from the aforementioned bridge linking effect that relies on the interfacial bonding of the U-nails.Both the geometric parameters of the U-nails and the geometric layout of the U-nail insertion have a substantial influence on the interlayer bonding behaviours. To derive the optimal design of U-nail reinforcement for 3D printed concrete, the effect of each factor on the mechanical properties is explored. Statistical analyses are carried out to determine the optimal design of the U-nail reinforcement.Analysis of variance is a statistical technique that is used to determine if the means of two or more groups of data are significantly different from each other. This method is often used to determine the influence of independent variables on the dependent variable in regression studies. The results of the variance analysis and F-test on the interlayer tensile and shear data are listed in , respectively, where the horizontal spacing interval and nail thickness are specified as influencing factors A and B, respectively. The F-values corresponding to both factors A and B are much greater than F0.05(4,9) = 4.26, indicating that both factors have a strong influence on the splitting tensile properties. The F-value for factor A is greater than that for factor B, indicating a stronger impact of the spacing distance than that of the thickness. For the intersection of factors A and B, the F-value is much lower than F0.05(4,17) = 2.96, demonstrating an insignificant interaction between the thickness and the spacing interval. Therefore, reducing the insertion spacing interval of nails is effective for improving the interlayer tensile bonding resistance.For interlayer shear bonding, the F-values corresponding to both factors A and B are greater than F0.05(4,9) = 4.26, indicating that the effects of both factors A and B are significant. In contrast to the tensile resistance, factors A and B exhibit similar effects on improving the interlayer shear bonding. Meanwhile, little interaction between the shearing capacities is indicated by the relatively small F-value for A × B. For the specimen subjected to shear action, reducing either the nail spacing interval or increasing the thickness is recommended to realise more effective reinforcement.RSM is a statistical method that is applied to solve practical problems by using a multiple quadratic regression equation to fit the functional relationship between the factors and response values. RSM considers the random error of the experiment. RSM is applied to evaluate the effects of the nail spacing interval and thickness on the interlayer tensile and shearing properties of 3D printed concrete specimens. presents the coefficients of determination, R2, and the R2 values adjusted according to different properties, which is an indication of the relationship between the independent variables (factors) and the response (selectivity). According to the calculated data, a linear model is appropriate for evaluating the splitting tensile behaviour, and a quadratic model is adopted for the shearing behaviour. shows the response surfaces and contours of the interlayer tensile strength with different U-nail arrangements. The results indicate that the tensile strength decreases gradually as both the U-nail spacing and thickness increase. This implies that the shear strength of the printed concrete can be improved by increasing the thickness of the U-shaped nails for a fixed spacing interval or by reducing the nail spacing interval. shows the response surfaces and contours of the interlayer shear strength with different U-nail layouts; the results indicate that the shear strength decreases gradually as the U-nail spacing increases. A greater layout spacing corresponds to a lower shear strength. For identical U-nail spacing, the shear strength initially increases and then decreases with increasing nail thickness. The optimal thickness is in the range of 2–2.5 mm.A method for depositing U-nails as vertical reinforcement concurrent with a 3D printing process is proposed to improve the interlayer bonding in this study. The vertically reinforced interfacial performance is investigated. The following conclusions can be drawn:The U-nail insertion device designed for 3D concrete printing coordinates well with the flexible and automatic construction process of 3DCP. In-process reinforcement for curved paths can be realized through real-time control of the rotation of the print nozzle.The improvement in the interlayer bonding depends on the layout of the U-nail reinforcements, i.e. the deposition space and depth, as well as the thickness of the nails. Notable reinforcement enhancement is achieved for the U-nail reinforced concrete composites compared with the unreinforced counterpart. The interface tensile strength increases with the addition of nails until the optimal addition for the interlayer shear capacity is reached.The interlayer tensile bonding strength is improved by 37.8%–61.8% through the bridging effect of the U-nail between adjacent layers. The reinforcement improvement of nail deposition is elucidated, and the pull-out work of the nails to resist interface de-bonding is calculated. The interlayer shear bonding strength is improved by up to 120% owing to the dowel action of U-nails to resist the interfacial shear stresses. A design method for U-nail deposition schemes to meet the desired shear capacity is presented.For tensile stresses, the incorporated U-nails exert a bridge linking effect on the interlayers through the friction forces with the cement matrix to restrict the splitting cracking, whereas for shear conditions, the U-nails perform a dowel function to improve the reinforcement. Based on statistical analyses, U-nails with a thickness of 2–2.5 mm are recommended to produce desirable improvements in the interlayer strength.The proposed solution is demonstrated to be effective in enhancing the interlayer bonding strength of 3DCP. Nail-reinforced concrete can be employed as permanent formworks for structural components and irregularly shaped panels. Nail-reinforced concrete is an effective supplement to reinforced concrete structures at the material level, but it is difficult to use nail-reinforced concrete to replace structural reinforcement systems. We believe that this method has promising applicability for construction-scale 3D printing. A mesoscale pull-out test and dowel action test of a single nail are necessary to provide an in-depth analysis of the nail reinforcement mechanisms. Further research efforts will be focused on improving the reinforcement effect in view of the physical and chemical interface processing of U-nails with high rigidity to improve the coordination between the nail and matrix. Meanwhile, more flexible adaptive deposition solutions that incorporate the topological optimisation design will also be explored.Li Wang: Conceptualization, Writing – original draft. Guowei Ma: Supervision, Writing – review & editing. Tianhao Liu: Investigation, Formal analysis. Richard Buswell: Writing – review & editing. Zhijian Li: Investigation.We have read and understood your journal's policies on copyright, ethics, etc., and believe that neither the manuscript nor the study violates any of them. We declare that the manuscript or any part of it has not been published, and is not under consideration for publication elsewhere in English or in any other language. We confirm that there is no conflict of interest.Evaluating creep cracking in welded fracture mechanics specimensAt present there lacks a unified approach in understanding the mechanistic role of weldments cracking in fracture mechanics specimens at elevated temperatures. The effects of residual stress, the development and crack tip damage due to it and the subsequent creep relaxation at high temperatures can be evaluated using relevant test data, numerical modelling and residual stress measurements. These problem areas are considered in this paper and discussed in context under the newly formed collaborative international effort in the Versailles Agreement of Materials and Standards committee, VAMAS TWA31. The plans for this collaborative effort are to evaluate tests at elevated temperatures in a number of high strength steels and also model and measure weldments containing residual stresses. The aim of the four year programme will be to make recommendations and establish pre-standardisation methods for testing, measuring and analysing creep crack initiation (CCI), creep crack growth (CCG), and low frequency creep fatigue crack growth (CFCG) (where creep dominates) characteristics in weldments containing residual stress. The fracture mechanics geometries that will be considered have already been validated in the previous TWA25 collaboration [VAMAS TWA25, Draft Code of Practice, ‘Creep/fatigue Crack Growth in Components’, VAMAS document. Nikbin K, editor. May 2005.] for testing of parent material. Examples of testing and analysis techniques are presented in this paper to highlight the future objectives for this work.The power generation industry is striving to meet criteria for clean and sustainable energy production by increasing efficiency while simultaneously decreasing levels of chemical emissions and pollutants. The efficiency of conventional steam and gas turbine power plant can be significantly improved by increasing the operating temperature, leading to reduced fuel consumption and lower levels of harmful emissions. Much uncertainty exists with regard to the treatment of residual stresses in component life prediction. Life assessment approach needs to incorporate innovative and advanced numerical modelling in conjunction with analytical and experimental techniques to ensure operational safety and efficiency of current and future conventional or nuclear plant. With the trend towards higher operating temperatures and the competing need to extend the life of existing power plant, more accurate and reliable experimental data for weldments are needed.Residual stresses invariably arise during fabrication and repair of components, particularly when components are joined by welding, and can occur during service (e.g. unplanned overloads). Post weld heat treatment (PWHT) can reduce the magnitude of residual stresses, but not completely remove them. Furthermore, PWHT is expensive and can lead to excessive distortion or material sensitisation. Current codes for design and assessment of high temperature components It is evident the important criteria for modelling and life assessment of weldments is first to develop an agreed testing procedure and analysis method. Improved accuracy in testing and analysis of data is an important factor in deriving the appropriate validated crack growth properties for both parent and weld related materials. A collaborative effort to identify methods in testing and analysis of weldments containing extreme inhomogeneity in material properties would assist in improving life assessment methods.The success of this collaborative effort will lie in delivering a unified view of the perspective of range of research interests in welding and weldments. This work will consider metallic alloy welds of interest to the power generation and chemical industries. Therefore in this integrated effort the first step is to identify important issues which are common to all the products. At the same time given the industries sensitivities of not wanting to divulge production and safety related issues care needs to be taken not to overstep the limits of confidentiality that industry relies on to keep their competitive edge.The goal for this feasibility study is to collect detailed breakdown of available data and consider common generic issues that all the partners need to solve in developing new techniques in weldments testing. This would consist of materials, weld methods, already available uniaxial and crack growth data on welds and where possible data on actual failures in welded components. The knowledge about material properties is essential for modelling and simulation of the component during the developmental, production and operational stages. Understanding weldments, combinations and effects of ‘ductile/brittle’ layers of weld/parent material, type and size of weld filler material, welding procedures and individual material properties of the various weld regions play an important part in improving modelling and life prediction. A data base of available core materials and materials relevant to the joining process and their available mechanical properties will be established. This will cover industrially relevant alloys used at elevated temperatures.There are available material characterisations data in previous EU collaborative projects The principal objectives from the VAMAS TWA31 collaboration will be to reach a mechanistic understanding of the interaction between residual stresses and primary loads and to quantify residual stresses and their effect on creep crack initiation and growth in components containing residual stress. A computational framework to model and analyse the interaction of primary loads, residual stresses and crack growth will be planned and the results will be validated by a Round Robin CCG testing of industrially relevant steel alloys and their weldments. These would include P22, P91, P92 steels and 316H, and 347 stainless steels. The information from these tests will be analysed and appropriate fracture mechanics parameters will be validated in their correlation. In addition the Round Robin modelling and residual stress measurements (using mainly neutron and X-ray diffraction is also planned in order to identify the important issues due to the presence of residual stress in welds and the subsequent stress redistribution due to high temperature testing. Procedures for treatment of residual stress in high temperature structural integrity assessments will be a deliverable to be integrated into the appropriate fracture mechanics parameters used in the analysis of the test results in weldments.Relevant parameters that will be considered and their relationship with crack initiation and growth have already been considered in a comprehensive manner for homogenous materials in TWA25 Crack growth in creep and fatigue can be described in various way using different correlating parameters. The correlations of steady-state crack growth rate a˙, with K, reference stress and C∗ can be represented by straight lines of different slopes on log/log plots and expressed by power laws of the formwhere A, Do, m and φ are material constants. A steady-state relationship between crack growth rate and the parameters in Eqs. and physically imply a progressively accelerating creep crack growth rate. The elastic stress intensity factor K and the C∗ parameter have generally been proposed for creep-brittle and creep-ductile materials, respectively. However it is necessary to verify the suitability of any of these parameters with respect to crack growth prediction in welded materials. For fracture mechanics specimens usually emphasis is placed on the analysis of CCG using C∗. This parameter which is derived from an analogy of J in a power-hardening material where ε˙0, σo, C and n are material constants at constant temperature. C∗ is obtained for fracture mechanics specimens experimentally in ASTM E1457 where P is the applied load, W |
− |
a is the remaining ligament ahead of the crack and Bn is the net thickness (= |
B for a specimen without side grooves) and F′ is a function of the materials creep properties and a geometric factor. The details of estimating F′ for different geometries are given in available publications Δ˙ is the measure load-line displacement rate consisting of the elastic, plastic and creep portions. In Eq. ideally the creep displacement rate. This assumes only a small amount of elastic and plastic components which is usually true for most creep ductile situations is calculated using the total measured displacement rate Δ˙ assuming Δ˙c≅Δ˙.The errors involved in this assumption are at most a factor of 2 for creep which falls well within the experimental scatter inherent in CCG data which have been found to be as much as a factor of 5 in any given one batch where a˙ is in mm/h, εf∗ is failure strain as a fraction and C∗ is in MJ/m2 |
h.εf∗ (as a fraction) is the creep ductility appropriate to the state of stress at the crack tip. For plane stress conditions εf∗ can be taken as the uniaxial ductility εf and for plane strain conditions as εf/30 can be an appropriate parameter to describe CCI and CCG rates in weldment.In a previous European collaborative programme tests on welded and parent P22 specimens of different size and geometries were carried out at 565 °C also bound the data. It is clear that within the scatter of the data the weldment tests consisting of HAZ do not show a difference in CCG rate with parent material. This could be due to the fact that P22 weldment has similar creep properties and microstructure to the parent material. However it is also possible that the C∗ needs to be estimated more accurately for the weldment test to highlight any CCG rate difference that may exist. These aspects will be investigated further in the Round Robin collaboration in TWA31.When a structure containing a defect is first loaded the stress distribution is given by the elastic K-field or the elastic–plastic J-field. Therefore, time is required for the stresses to redistribute to the steady-state creep stress distribution controlled by C∗. In addition, a period of time is needed for creep damage to develop around the crack tip In this procedure, it has been determined that time to Δa |
= 0.5 mm is a suitable value to adopt as an ‘engineering’ limit for CCI in fracture mechanics geometries. From Eq. it may be expected that the time, ti, to initiate a crack extension of Δa can be expressed as giving:where Di and ϕi are material constants. For steady-state cracking Di is expected to be given approximately by da/D withThese equations assume that the entire initiation period is governed by steady-state C∗. This cannot be expected to be true during at least part of the initiation period ti. For the CCI correlation, the time to 0.5 mm crack growth versus C∗ or K is usually plotted and for CCG correlation a plot of crack growth rate, a˙, as a function of C∗ or K is plotted after the appropriate parameter has been validated can be approximated and bound in the same manner as Eq. Alternatively if the incubation period is calculated from the initial transient cracking rate a˙o determined from the approximate upper-bound tiU to the initiation time becomes the incubation period is proportional to da. The limit of reliable crack detection is at best ±100 μm (which sets a level of da |
= 0.2 mm for standard CT testing in ASTM E1457 the time to crack extension of da |
= 0.5 mm versus the value of C∗ at the end of crack length a |
+ da. The scatter is normal in the correlation and there is no clear difference between weldment tests and parent material. Predictions from Eqs. bound the data adequately. Where weldment tests and residual stresses are concerned the initiation period is an important regime to observe as the crack tip stresses would have had less time to redistribute during the early stages. It is expected that the differences with weld materials would show the largest effect in creep/brittle conditions. These aspects will be further investigated in the Round Robin testing.Although the main thrust of the work is related to time-dependant cracking either under static loading or low frequency (<0.1 Hz) cyclic loading) it is useful to refer the simple methods available in codes for cases where there is a combination of effects due to creep and fatigue. For fatigue crack growth it is assumed that the mechanism is time and temperature independent and K or J dominates at the crack tip where da/dN is fatigue crack growth rate per cycle, C′ and m′ are material dependent parameters, which may be sensitive to the minimum to maximum load ratio R of the cycle. It is also well known that the Codes of Practice At elevated temperatures combined creep and fatigue crack growth may take place. However in most cases fatigue dominates at higher frequencies (f |
> 1 Hz) and creep dominates at lower frequencies and dwell periods (f |
< 0.1 Hz) where this linear summation combines creep and creep/fatigue components. This method can be refined using the method given in the British Energy’s R5 Procedure . The simple cumulative damage law can be applied to describe creep/fatigue interactions . These aspects will be discussed further in this paper.As presented above when making measurements of the creep crack growth properties of materials according to the ASTM E 1457 where h is a non-dimensional function of crack size, component dimensions, state of stress and n. Also, as ε˙ is instantaneous creep strain rate, Eq. can be employed to incorporate primary, secondary and tertiary creep when calculating C∗. When secondary creep prevails and creep strain rate ε˙ can be expressed in terms of stress σ by a relation of the formwhere ε˙0, σ0, C and n are material constants then Eq. is of exactly the same form as the expression used in plasticity for JFor the same loading conditions, crack geometry, component dimensions and when np |
= |
n, the same stress distribution is obtained ahead of a crack in a creeping material as in a work-hardening material and h will be the same in both cases so that C∗ can be determined from Eqs. enables estimates of C∗ to be obtained when solutions for J only are available An alternative procedure for estimating C∗ is to use limit analysis methods where PLC is the collapse load of the uncracked ligament for a material of yield stress σY. Solutions for rσref for a number of applications are included in is chosen as the reference stress, that h in these equations becomes relatively insensitive to n or np. This means that it can be evaluated from G, for which np |
= 1 to givewhen G is expressed in terms of K. This equation provides estimates of C∗ from solutions of K and σref that are available in the codes of practice In the above relationships no allowance has been included so far to include in the analyses the possible presence of residual stress in a test specimen or a component. A characteristic of a residual stress distribution is that it must satisfy force and moment equilibrium. It will not, therefore, affect collapse and the value of reference stress σref. However, a tensile residual stress distribution at a crack tip will tend to enhance early crack growth and a compressive stress distribution retards it . As a consequence it is likely to affect the initial transient phase of cracking and the duration of an incubation period.An approximate procedure for allowing for the influence of residual stress is to calculate a combined stress intensity factor K due to the applied loading Kapp and that due to the residual stress distribution Kres such thatThis enables a conservative estimate of the transition time to achieve widespread creep conditions at a crack tip. ASTM E1457 where C∗ in this expression is its value once widespread creep conditions have been reached at the crack tip. It will be equal to the value from the applied loading only due to relaxation of the residual stress field. It can be obtained by substituting K |
= |
Kapp in Eq. , the presence of tensile residual stress at a crack tip will increase the duration of tτ. For t |
< |
tτ the stress local to a crack tip will be higher than predicted from using the steady-state value of C∗ in Eq. . This can be compensated for by calculating C∗ for this period, when determining crack growth rates and incubation periods, fromIt is expected that application of this formula will be conservative as Kres will relax by creep towards zero as t approaches tτ.A key feature of using this method is to have accurate estimates of residual stress distributions available so that reliable determinations of Kres can be obtained. Several procedures are available for measuring residual stress distributions in components For the design and initial development procedures of any component there are substantial number of computational tools available. The common link between the designing of all joining and welded products is that they tend to be invariably the weakest part within the structure. The integration of advanced modelling design and simulation, which is common to all components, could drastically reduce costs and improve efficiency. In this respect the analysis for residual stress and the models applied will be considered in detail and validations of the results will need to be performed to increase confidence.An important issue in all joining is the presence of cracks or microvoids or fracture in the weld regions of the components and therefore modelling cracked components with residual stress will need to be performed. Fracture mechanics parameters are used to characterise crack initiation and growth at high temperatures. Typically, at short times the stress intensity factor K is used and at long times, the non-linear parameter C∗ is used. Under higher loads and at short times, failure can be characterised using the J integral and methods to incorporate residual stress into the driving force for fracture J have been developed The initial generation of the residual stress fields will generally be associated with the introduction of microscale damage, which must be accounted for in any lifetime prediction. Due to stress redistribution during creep, the elastic strains associated with the residual stress field are converted to inelastic (creep) strains over long times as the stresses relax, eventually towards zero. Furthermore, for the general case of a secondary (residual) stress the secondary stresses are treated as primary in the estimation of the stress intensity factor used to predict creep crack growth.Following welding procedures there are invariably residual stresses induced in the interface region. Actual numerical simulation of welds can be performed to obtain residual stress profiles. The results from this method are not easily comparable with an actual weldment as there are a number of variables and assumptions that would affect the results. Another way is to induce residual stress in a controlled manner using a plastic overload in a geometry containing a sharp or blunt crack. In this way both numerical and residual stress measurements can be calculated and compared.Previous modelling and residual stress measurement on crack geometries have been carried out on C-ring specimens a shows the example of the one half of the C-ring mesh which goes through the stage of loading and unloading as represented by the hoop stress contours shown in shows the results of the hoop stress distribution for P22, P91 and type 316 LN stainless steel after unloading. Although the trends are the same the figure indicates that the peaks are directly dependent of the yield stress of the alloy. Also it can be interpreted that the more ductile type 316 LN stainless steel (around 60% ductility) gives the lowest stress levels and the more brittle P91 (around 20% ductility) give the highest peaks. shows the comparison between finite element results using elastic-plastic analysis and actual measurements in P91 steel. Clearly it is possible to identify and predict trends using FEM analysis but comparison of actual need improvement. It is planned to perform a Round Robin Modelling and measurements of a Compact Tension specimen with brittle properties. Preliminary numerical analyses Three candidate materials that have been chosen for simulating residual stresses in a CT shape specimen consist of a brittle weld 347 stainless steel a. The finite element mesh to analyse the specimen is similar to recent work on the geometry b shows an example of the plastic zone created after overload. shows the stress distribution normal to the crack plane for a 3D FEM run Creep relaxation is likely to play an important part in reducing the effects of tensile residual stress on CCG rates. This can be modelled using the overload model shown in shows the reduction in normal stresses with time following heating at 650 °C. The stresses are again normalised by the room temperature 0.2% flow stress. It is seen that the peak normal stress reduces by almost 50% from time zero to 1000 h. The subsequent drop at time = 10,000 h is very much less. Similar reduction is seen for the other stress components It should be noted that the effects due to crack tip stresses alone may not be the only reason for the CCG rate to be affected Following the development of a unified weldment testing and methodology a detailed understanding of the behaviour of the weld product at the operational stage undergoing sustained static or cyclic loads and in corrosive and elevated temperature environments is essential in order to validate the findings. The improvement of present industrial practice in this respect will mean understanding the testing and examination procedures and identifying the numerical and analytical models that are available as future tools for lifing methodologies. A survey of weld component failures which are available in the public domain will assist in the further understanding of the importance of using laboratory weldment test data in life assessment methods.There is without doubt a considerable lack of scientific evidence to support the current assumptions in assessing the life of high temperature components that contain residual stress. This stems from a lack of verifiable and reliable fracture mechanics properties data on weldment testing and analysis and the difficulties faced in the correct modelling and measurement of residual stresses in weldments. The challenge for the VAMAS TWA31 collaborative program is to identify the conditions under which residual stress effects are important at high temperatures and devise methods for testing and analysis to incorporate such effects into structural integrity procedures for prediction of crack initiation and growth.In this paper the different correlating parameters that might be appropriate to weldment testing have been discussed in brief. Methods of calculating C∗ both experimentally and in conjunction with uniaxial creep data for components have been presented. It is demonstrated how this term can be evaluated from the plastic fracture mechanics parameter J or, approximately, from stress intensity factor K and limit analysis using reference stress σref concepts. Procedures for dealing with a crack initiation period and steady state during which damage builds up at a crack tip have been considered. The role of residual stress on this period and subsequent crack growth has also been examined. It is clear from the findings that there is scope within TWA31 to improve the testing and analysis of weldments specimens.The plan therefore is to propose methods to derive validated experimental creep and creep/fatigue crack growth properties for weldments material. To this end the main objectives for TWA31 have been identified to be as follows:A survey and a collection of a database of weld and weldments testing which will include the specimen categorisation in terms of the size of weld region, ‘ductile/brittle’ weldments properties and welding methods. Collaboration with other committees such as ASTM E08 creep crack growth committee and the EU ESIS TC11 WG:HTTW weldment test committee to share information and experience is an essential part of this task.Perform a Round Robin residual stress measurements of weld fracture mechanics tests of a number of industrially used steels e.g. P22, P91, 316H and P92 steels using neutron, X-ray and deep hole drilling techniques.Perform numerical modelling of fracture mechanics specimen to induce residual stresses to model weld residual stress profiles and quantifying constraint effects due to weldments and subsequently validating the results by means of residual stress measurements that will be carried out.Perform a Round Robin testing and analysis of weld fracture mechanics tests of a number of industrially used steels e.g. P22, P91, 316H and P92 steels.Validating the appropriate fracture mechanics parameters in weld/HAZ/parent type specimens and considering the effects of residual stresses and their subsequent relaxation during high temperature operations.Finally by using the information derived from 1 to 5 to apply the results to life assessment methods and to propose pre-standard recommendations in weldments testing and analysis techniques.Analysis and mitigation of seismic pounding of a slender R/C bell towerPounding is one of the greatest sources of seismic vulnerability of slender R/C structures, including civic or bell towers. An emblematic case study falling in this class of structures, i.e. a modern heritage R/C bell tower constructed in the early 1960s to replace the former 19th century tower of the Chiesa del Sacro Cuore in Florence, is analyzed in this paper. In order to assess the effects of pounding, a special multi-link viscoelastic finite element contact model was devised and calibrated in this study to reproduce Jankowski’s non-linear viscoelastic analytical model. Indeed, the damping coefficient of the latter is defined as a non-linear function of time, and thus it cannot be directly implemented in commercial calculus programs, because the damping coefficient of the damper elements included in their basic libraries is assumed to be a constant. The non-linear dynamic enquiry carried out by the finite element model of the bell tower and the church incorporating the multi-link viscoelastic contact elements shows that pounding affects the seismic response of the two buildings as early as an input seismic action scaled at the amplitude of the normative basic design earthquake level. Furthermore, unsafe stress states are highlighted for the columns of the tower under seismic action scaled at the maximum considered earthquake level. A damped-interconnection retrofit solution consisting in linking the two structures by means of a pair of pressurized fluid-viscous dissipaters is proposed to prevent pounding. The technical implementation details of this rehabilitation strategy are illustrated, and the benefits induced are discussed by comparison with the response in original conditions and in the hypothesis of a conventional rigid-connection retrofit intervention between the bell tower and the church.A significant stock of modern heritage buildings is constituted by reinforced concrete (R/C) structures designed and erected with no—or very limited—seismic provisions, due to the lack of reference Technical Standards at the time of their construction In this respect, the case study analyzed in this paper is emblematic, being represented by a R/C bell tower constructed in the early 1960s to replace the existing 19th century tower of the Chiesa del Sacro Cuore in Florence. The construction of the new tower was part of a comprehensive intervention of refurbishment and enlargement of the church carried out during the same period. The architectural and structural designers were two well-known Florentine professionals of the time, Lando Bartoli and Lisindo Baldassini, respectively. The world-famous structural engineer Pier Luigi Nervi co-operated to design the bracing system of the new bell tower. In order to emphasize its monumental and religious value, the tower was conceived as a slender nude reticular R/C structure simulating the shape of two hands joined in prayer. Moreover, it was erected in front of the façade of the church, rather than to its right back side, where the former bell tower used to be found. This causes the new tower to be totally visible from all standpoints of a long straight residential street at one end of which the church is situated. The space occupied by the demolished bell tower was used to add a new wider apse, covered by an octagonal R/C dome. Two photographic views of the church, showing its appearance in the late 1950s, before the enlargement interventions, and in its current configuration, are displayed in The new tower was built at a very narrow distance from the façade of the church, plastered with artistic marble sheets. The separation gap corresponds to the thickness of the wooden formworks used to cast the constituting R/C members, plus a thin cardboard sheet inserted to protect the façade during casting. The thickness of the gap slightly varies along the height, due to small local discontinuities of the formworks, and reaches a minimum of about 20 mm in correspondence with some potential impact spots situated on the rear arcade beam of the tower. This is witnessed by the photographic images taken during the works reproduced in . As a consequence of the little width of the gap, the two structures appear to be remarkably pounding-prone. This is also highlighted by their modal properties, which are discussed in the next section.In order to assess the effects of pounding, a non-linear dynamic enquiry was carried out by simulating collisions by means of a special finite element multi-link viscoelastic contact model, originally implemented and calibrated in this study to reproduce the behavior of Jankowski’s non-linear viscoelastic analytical model Based on these data, a retrofit hypothesis is proposed according to the most recent strategies in pounding mitigation, based on the “damped-interconnection” concept , the structure of the bell tower is constituted by two twin braced R/C frames, each being supported by four columns with variable sections along the height. The shape of the tower mirrors the geometry of the church nave portion up to the top of the façade. This allows creating a narrow arcade, completed by two smaller side frames with the same shape as the aisle portions of the façade. The roof of the arcade represents the geometrical outer continuation of the church roof, and is made of a R/C slab ribbed by the arcade beams of the two main frames, plus a central beam stemming from the lateral frames. The slab, an intrados view of which is offered in the left image of , is the only connection between the eight main columns from the foundation—made of a mesh of inverted T-beams—to the arcade roof. Over the roof, the columns are linked by six “star”-shaped horizontal plates situated at four different levels. These plates also connect the eight pairs of diagonal R/C beams constituting the flights of stairs provided to reach the belfry, which also represent the bracing system of the tower frames in the direction parallel to the façade (right image in ). The four external columns merge in a robust slab situated at the bottom of the belfry, whereas the four internal columns cross this slab and continue up to the top of the belfry. This final portion of the internal columns takes a trapezoidal shape up to the top of the tower, so as to provide visual continuity with the geometry of the external columns. The imposing dimensions, the futuristic look, and the architectural and structural quality of the tower has conferred it the role of a modern heritage building in the Florentine landscape over the past decades.The refurbishment and expansion works carried out in the church before the construction of the bell tower remarkably transformed both its architectural appearance and structural characteristics. Indeed, the only portions of the original building preserved after the works were the longitudinal masonry walls of the aisles, the glazed walls situated over the colonnades of the nave, and the wooden trusses of the nave roof. The masonry columns of the two colonnades were substituted with new R/C members, and the wooden roofs of the aisles with R/C slabs. As mentioned in the introduction, a new wider apse was added, whose structure is constituted by four R/C frames supporting an octagonal R/C dome, totally covered with decorated glass panes along its perimeter.An accurate 3D geometrical model of the church and the bell tower, illustrated in the left image of , was developed based on the original design documentation collected through record research, as well as on supplementary field laser measurements. The mechanical properties of concrete and steel and the reinforcement details were also drawn from the original design drawings, as well as from the calculus and technical reports of the new bell tower structure. These reports also contain information on the characteristics of the materials constituting the wooden trusses and the masonry walls of the church. These documents highlight, for concrete, a characteristic value of compressive strength, fck, equal to 25 MPa, and a Young modulus, Ec, of 31,000 MPa; and for reinforcing steel, a yield stress, fy, equal to 450 MPa, and a Young modulus, Es, of 206,000 MPa. The stone masonry has a regular texture, with the following estimated properties: compressive strength, fmk, of 5 MPa; tensile and shear strength for null compression, ftk and fvk, of 0.15 MPa; and a Young modulus, Em, of 3000 MPa. The wood of the trusses has characteristic bending, compressive and tensile strengths, fwbk, fwck and fwtk, equal to 24, 21 and 11 MPa, respectively, and a Young modulus in the direction parallel to the wood fibers, Ew0, of 11,000 MPa. These data were transferred into the finite element models of the two structures generated by the SAP2000NL calculus program The results of the modal analysis carried out by the finite element models of the two structures show a first mode of the tower alone, mixed translational along the direction orthogonal to the façade (named y in geometrical model drawing of )–rotational around the vertical axis (z), with vibration period of 1.94 s and effective associated masses equal to 80% of the seismic mass of the tower along y and 22.4% around z. The corresponding shape is plotted in the right image of . The second mode, concerning the bell tower alone too, is mixed translational along the direction parallel to the façade (x)-rotational, with period of 1.39 s and associated masses equal to 93.1% along x, and 66.9% around z. The third mode is translational along x-rotational, with period of 0.39 s and associated masses equal to 3.4% along x, and 2.7% around z. The fourth mode is translational along y-rotational, with period of 0.38 s and associated masses equal to 14.9% along y, and 3.9% around z. These first four modes activate a summed effective modal mass equal to about 95% of the seismic mass of the tower for all three reference axes, that is, greater than the 85% value representing the minimum percentile fraction required by the Italian Technical Standards The church structure features all mixed translational–rotational modes too, of which the former has a period of 0.48 s. The first modes that include a significant translational contribution in y direction are the fifth and sixth ones, with vibration periods of 0.21 s and 0.17 s, equal to about 1/9 and 1/11 of the first period of the tower along the same axis. This highlights very different dynamic properties of the two structures along the potential pounding direction, as expected from their structural characteristics. In total, 20 modes are needed for the church to activate a summed modal mass greater than 85% of its total seismic mass.The finite element analysis of pounding between bell tower and church was carried out according to a classical “contact element approach”, which offers a straightforward idealization of the problem, as it corresponds to the intuitive interpretation of the phenomenon. Impact is simulated by a contact element that is activated when the relative displacement u1(t) − |
u2(t) between the two structures—u1(t), u2(t) being the displacements of the first and second structure, respectively, both functions of time, t—equals the width of the existing gap at rest, gr. This allows solving the problem within the framework of an ordinary time-history response analysis where v1(t), v2(t) are the approaching velocities of the two structures, and v1′(t), v2′(t) are the post-impact (rebound) velocities. In earlier finite element pounding computations accounting for the energy loss that occurs during collision, the contact element has been modeled by the classical Kelvin–Voight linear viscoelastic rheological scheme, i.e. combining in parallel an elastic spring, which is capable of transmitting the impact forces, and a linear viscous damper, which simulates the impact-related energy dissipation In later studies, a gap (no-tension) element has been incorporated in series with the damper, so that the latter is activated at the approaching stage of the colliding structures only, rather than in the rebound phase too (i.e. when relative velocity becomes negative). Unlike the original Kelvin–Voigt model, this avoids spurious energy dissipation during bounce and provides a more accurate physical interpretation of pounding. This improved model corresponds to the analytical formulation proposed in , where the structures are idealized as rigid masses, indicated with symbols m1 and m2, the impact force-transmitting spring is denoted by its axial stiffness kH, the existing separation gap at rest is named gapr, and the gap element that disconnects the damper in the restitution phase gapc. Moreover, in this scheme the additional elastic spring with stiffness kd placed in parallel with the damper is aimed at driving it to its pre-impact position before a new contact occurs. By developing a solution formulated in For pounding computation, the n exponent is fixed at 3/2 where β is the impact stiffness parameter of the spring, which has the dimensions of a force divided by a 3/2-power law of displacement. In model where the impact damping ratio ξ is expressed asBased on relations (1) through (5), the total viscoelastic non-linear contact force, Ft, results to be:Ft(t)=β·δ(t)3/2+cnl(t)·δ̇(t)forδ(t)>0andδ̇(t)>0(contact-approach phase)Ft(t)=β·δ(t)3/2forδ(t)>0andδ̇(t)⩽0(contact-restitution phase)Ft(t)=0forδ(t)⩽0(no contact)According to these non-linear behavioral hypotheses, the kH stiffness of the impact force-transmitting spring is regarded as an effective impact stiffness, given by the following function of δ(t):The non-linear viscoelastic model defined by relations (4)-(7) represents a reasonable balance point between the need to reach accurate results in the reproduction of structural pounding and to limit computational effort In order to overcome this limitation, a special “multi-link viscoelastic” model constituted by an in-series assemblage of m linear dampers and m associated in-parallel linear springs was devised in this study. The response of the model is based on the sequential activation and disconnection of the dampers, following the variation of the interpenetration depth. This way, the resulting equivalent damping coefficient of the assemblage becomes a function of δ(t), and expression (4)—as well as any other relation between cnl and δ(t) likely to be selected in the analysis, such as the ones proposed in The version of the multi-link viscoelastic model with m |
= 5 components is drawn in the right image of , where the dampers are denoted by relevant damping coefficients ci (with i |
= 1, …, 5 in this case, and i |
= 1, …, |
m in general). The activation of each damper is governed by a gap (named gapci in ), to which a prefixed opening, wi, is assigned. As the gap closes, the damper starts to react, adding its response to the already activated dampers. Like for the non-linear damper of the rheological scheme in the left image of , each element is combined with a linear spring, with stiffness kdi, which drives it to its pre-impact position. The remaining components of the assembly (non-linear Hertzian spring, the stiffness of which is denoted by symbol kHFE in the finite element assembly, and the gap at rest gapr) are the same as in model The calibration of the initial openings wi of the gapci elements and the ci damping coefficients of the dampers can be carried out according to the two-step procedure summarized below.Estimation of wi values. This is related to the tentatively predicted value of the maximum interpenetration depth, δmax, that will be achieved in the time-history analysis of the colliding buildings. A numerical enquiry carried out on low-to-medium rise R/C frame structures with total heights up to 25 m – a demonstrative case study of which is presented in Calculation of ci values. As observed above, due to the in-parallel assembly of the dampers, the damping contribution of each one of them is numerically added to the contribution of the already activated dampers. Therefore, the ci values must be assigned in such a way that the resulting “summed” damping coefficient of all the elements activated in correspondence with a given δ¯ value, cs(δ¯), satisfactorily matches the corresponding analytical cnl(δ¯) value computed by (4). Then, the fitting process of cs to cnl is carried out by imposing that the value of the “summed” damping coefficient reached at the activation of the i-th damper, csi, is equal to the cnl coefficient calculated by (4) for the value of the interpenetration depth situated in the middle of the interval of activation of the same damper, δi, i.e. csi |
= |
cnl(δi), with δi=wi+Δ2. Afterwards, ci is obtained by subtracting the values of the damping coefficients of the dampers activated before the ith damper from the csi summed value: ci=csi-∑j=1i-1cj. This generates the following sequence of ci values: c1 |
= |
cs1; c2 |
= |
cs2 |
− |
c1; …; |
cm=csm-∑j=1m-1cj.The last set of parameters to be fixed for the multi-link model is represented by the kdi stiffness values of the linear springs associated in parallel to the dampers. In this case, an empirical criterion is followed for their preliminary estimation, which consists, as observed above, in calibrating the kdi values in order to allow the springs to disconnect the dampers completely (i.e. reopen relevant gaps) in the restitution phase, before a new contact occurs. This calibration process could require one or two iterations, starting from an initial set of tentative stiffness values. The analyses carried out on the R/C frame structures mentioned at point 1 above suggest adopting for the kdi values the same mutual proportions existing among the ci coefficients calculated according to point 2. Following this empirical criterion, the tentative choice of the spring stiffness values is reduced only to kd1. For the numerical test R/C structures examined, kd1 resulted to vary from about 1000 kN/m to about 2000 kN/m. Thus, to properly start the iterative search process of the set of kdi values, it is recommended to select kd1 within this range, and then to impose the relations: kd2=c2c1·kd1; |
kd3=c3c2·kd2; …; |
kdm=cmcm-1·kdm-1, for kd2 through kdm. To duplicate exactly the kH values of the analytical model given by (7), the kHFE stiffness values of the impact force-transmitting spring of the multi-link model should be corrected in the m intervals of activation of the dampers so as to take into account the additional kdi contributions (normally in the order of few percent units of kHFE). This is obtained by simply subtracting the sum of the kdi values of the springs activated in a given interval from the kHFE values computed from (7) for the same interval.A numerical enquiry was carried out prior to develop the time-history analysis of pounding between the bell tower and the church, where the m number of damper + gap + spring elements of the multi-link viscoelastic model was varied from 3 to 9, so as to evaluate the relevant influence on the reproduction of relations (6) for this case study. Of the seven models above, the 5-link version shown in proved to bear the best balance between numerical simulation performance, which resulted to be very similar for m ranging from 5 to 9 (slightly poorer for m |
= 3 and m |
= 4), and computational effort, processing times being approximately an exponential function of the number of damper + gap + spring elements. Based on these data, the 5-link model was finally selected for the dynamic analysis of pounding.The contact elements were included in the positions marked by the pairs of joints denoted by letters A–A′ through E–E′ in , which represent the most likely potential impact spots situated on the four rear columns and at the top of the rear arcade beam of the tower, as observed in the Introduction, and the corresponding spots on the church façade. Indeed, due to the shape of the formworks used for the concrete cast, the internal side of the rear rib beam of the arcade reaches the local minimum distance of 20 mm from the façade in the zones situated across these five potential spots.The characteristic parameters of the model were fixed as follows. A 0.65 value of the coefficient of restitution e, typically adopted for concrete-to-concrete impact The colliding masses were computed by considering the weight of the portion of the bell tower situated over the arcade (weight of the arcade included), for m1, and the weight of the church roof plus the weight of the supporting walls of the roof situated over the colonnades of the nave, for m2. The resulting mass values were divided in equal fractions among the 5 pairs of contact joints, providing the following values of m1 (tower) and m2 (church) for each pair: m1 |
= 84.5 kN s2/m, m2 |
= 86.5 kN s2/m.The gr opening was set as equal to the minimum width of the physical gap, that is 20 mm, for all 5 pairs of contact joints. The gap openings and damping coefficients of the five dampers of the 5-link model were fixed according to the criteria illustrated in points 1 and 2 of Section . Considering that the impact spots A–A′ through E–E′ are situated at heights comprised between about 19 m and about 21 m, δmax is expected to range from about 5 mm (2.5 × 10−4 times of the rounded average height) to about 10 mm (5 × 10−4 times). The mean value of this penetration depth interval, i.e. 7.5 mm, is consequently assumed as tentative estimate δmax,t of the maximum interpenetration depth. Then, the width of the 5 intervals of sequential activation of the dampers results to be: Δ=δmax,t5=1.5mm, and thus the gap openings: w1 |
= 0; w2=Δ=1.5mm; w3=2Δ=3mm; w4=3Δ=4.5mm; and w5=4Δ=6mm.Based on these wi values, the δi=wi+Δ2 mean interpenetration depths of the five intervals are as follows: δ1 |
= 0.75 mm; δ2 |
= 2.25 mm; δ3 |
= 3.75 mm; δ4 |
= 5.25 mm; and δ5 |
= 6.75 mm. The “summed” damping coefficients csi |
= |
cnl(δi) calculated by substituting in (4) the ξ, β, m1, m2, and δ1 through δ5 values above amount to: cs1 |
= 1325 kN s/m; cs2 |
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